chapter 6 resource masters - algebra 1 · 2018-10-17 · v assessment options the assessment...

Chapter 6 Resource Masters

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Consumable Workbooks Many of the worksheets contained in the Chapter Resource Masters are available as consumable workbooks in both English and Spanish.

ISBN10 ISBN13Study Guide and Intervention Workbook 0-07-890848-5 978-0-07-890848-4Homework Practice Workbook 0-07-890849-3 978-0-07-890849-1

Spanish VersionHomework Practice Workbook 0-07-890853-1 978-0-07-890853-8

Answers for Workbooks The answers for Chapter 6 of these workbooks can be found in the back of this Chapter Resource Masters booklet.

StudentWorks PlusTM This CD-ROM includes the entire Student Edition test along with the English workbooks listed above.

TeacherWorks PlusTM All of the materials found in this booklet are included for viewing, printing, and editing in this CD-ROM.

Spanish Assessment Masters (ISBN10: 0-07-890856-6, ISBN13: 978-0-07-890856-9)These masters contain a Spanish version of Chapter 6 Test Form 2A and Form 2C.

Copyright © by the McGraw-Hill Companies, Inc. All rights reserved. Permission is granted to reproduce the material contained herein on the condition that such material be reproduced only for classroom use; be provided to students, teachers, and families without charge; and be used solely in conjunction with Glencoe Geometry program. Any other reproduction, for sale or other use, is expressly prohibited.

Send all inquiries to:Glencoe/McGraw-Hill8787 Orion PlaceColumbus, OH 43240 - 4027

ISBN13: 978-0-07-890515-5ISBN10: 0-07-890515-X

Printed in the United States of America.

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iii

Teacher’s Guide to Using the Chapter 6 Resource Masters .........................................iv

Chapter ResourcesStudent-Built Glossary ....................................... 1Anticipation Guide (English) .............................. 3Anticipation Guide (Spanish) ............................. 4

Lesson 6-1Angles of PolygonsStudy Guide and Intervention ............................ 5Skills Practice .................................................... 7Practice .............................................................. 8Word Problem Practice ..................................... 9Enrichment ...................................................... 10

Lesson 6-2ParallelogramsStudy Guide and Intervention .......................... 11Skills Practice .................................................. 13Practice ............................................................ 14

Word Problem Practice ................................... 15Enrichment ...................................................... 16

Lesson 6-3Tests for ParallelogramsStudy Guide and Intervention .......................... 17Skills Practice .................................................. 19Practice ............................................................ 20Word Problem Practice ................................... 21Enrichment ...................................................... 22

Lesson 6-4RectanglesStudy Guide and Intervention .......................... 23Skills Practice .................................................. 25Practice ............................................................ 26Word Problem Practice ................................... 27Enrichment ...................................................... 28TI-Nspire Activity ............................................. 29Geometer’s Sketchpad Activity ....................... 30

Lesson 6-5Rhombi and SquaresStudy Guide and Intervention .......................... 31Skills Practice .................................................. 33Practice ............................................................ 34Word Problem Practice ................................... 35Enrichment ...................................................... 36

Lesson 6-6Trapezoids and KitesStudy Guide and Intervention .......................... 37Skills Practice .................................................. 39Practice ............................................................ 40Word Problem Practice ................................... 41Enrichment ...................................................... 42

AssessmentStudent Recording Sheet ................................ 43Rubric for Extended Response ....................... 44Chapter 6 Quizzes 1 and 2 ............................. 45Chapter 6 Quizzes 3 and 4 ............................. 46

Chapter 6 Mid-Chapter Test ............................ 47Chapter 6 Vocabulary Test ............................. 48Chapter 6 Test, Form 1 ................................... 49Chapter 6 Test, Form 2A ................................. 51Chapter 6 Test, Form 2B ................................. 53Chapter 6 Test, Form 2C ................................ 55Chapter 6 Test, Form 2D ................................ 57Chapter 6 Test, Form 3 ................................... 59Chapter 6 Extended-Response Test ............... 61Standardized Test Practice ............................. 62Unit 2 Test ....................................................... 65

Answers ......................................... A24–A34

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Teacher’s Guide to Using the Chapter 6 Resource Masters

The Chapter 6 Resource Masters includes the core materials needed for Chapter 6. These materials include worksheets, extensions, and assessment options. The answers for these pages appear at the back of this booklet.

All of the materials found in this booklet are included for viewing and printing on the TeacherWorks PlusTM CD-ROM.

Chapter ResourcesStudent-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to recording definitions and/or examples for each term. You may suggest that student highlight or star the terms with which they are not familiar. Give to students before beginning Lesson 6-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.

Anticipation Guide (pages 3–4) This mas-ter presented in both English and Spanish is a survey used before beginning the chapter to pinpoint what students may or may not know about the concepts in the chapter. Students will revisit this survey after they complete the chapter to see if their percep-tions have changed.

Lesson ResourcesStudy Guide and Intervention These masters provide vocabulary, key concepts, additional worked-out examples and Check Your Progress exercises to use as a reteaching activity. It can also be used in conjunction with the Student Edition as an instructional tool for students who have been absent.

Skills Practice This master focuses more on the computational nature of the lesson. Use as an additional practice option or as homework for second-day teaching of the lesson.

Practice This master closely follows the types of problems found in the Exercises sec-tion of the Student Edition and includes word problems. Use as an additional prac-tice option or as homework for second-day teaching of the lesson.

Word Problem Practice This master includes additional practice in solving word problems that apply the concepts of the lesson. Use as an additional practice or as homework for second-day teaching of the lesson.

Enrichment These activities may extend the concepts of the lesson, offer a historical or multicultural look at the concepts, or widen students’ perspectives on the math-ematics they are learning. They are written for use with all levels of students.

Graphing Calculator or Spreadsheet Activities These activities present ways in which technology can be used with the con-cepts in some lessons of this chapter. Use as an alternative approach to some concepts or as an integral part of your lesson presenta-tion.

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Assessment OptionsThe assessment masters in the Chapter 6 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assess-ment.

Student Recording Sheet This master corresponds with the standardized test practice at the end of the chapter.

Extended-Response Rubric This master provides information for teachers and students on how to assess performance on open-ended questions.

Quizzes Four free-response quizzes offer assessment at appropriate intervals in the chapter.

Mid-Chapter Test This 1-page test pro-vides an option to assess the first half of the chapter. It parallels the timing of the Mid-Chapter Quiz in the Student Edition and includes both multiple-choice and free-response questions.

Vocabulary Test This test is suitable for all students. It includes a list of vocabulary words and 10 questions to assess students’ knowledge of those words. This can also be used in conjunction with one of the leveled chapter tests.

Leveled Chapter Tests

• Form 1 contains multiple-choice questions and is intended for use with below grade level students.

• Forms 2A and 2B contain multiple-choice questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Forms 2C and 2D contain free-response questions aimed at on grade level students. These tests are similar in format to offer comparable testing situations.

• Form 3 is a free-response test for use with above grade level students.

All of the above mentioned tests include a free-response Bonus question.

Extended-Response Test Performance assessment tasks are suitable for all students. Samples answers and a scoring rubric are included for evaluation.

Standardized Test Practice These three pages are cumulative in nature. It includes three parts: multiple-choice questions with bubble-in answer format, griddable questions with answer grids, and short-answer free-response questions.

Answers• The answers for the Anticipation Guide

and Lesson Resources are provided as reduced pages.

• Full-size answer keys are provided for the assessment masters.

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Chapter 6 1 Glencoe Geometry

Student-Built Glossary

This is an alphabetical list of the key vocabulary terms you will learn in Chapter 6. As you study the chapter, complete each term’s definition or description. Remember to add the page number where you found the term. Add these pages to your Geometry Study Notebook to review vocabulary at the end of the chapter.

6

Vocabulary TermFound

on PageDefinition/Description/Example

base

diagonal

isosceles trapezoid

legs

midsegment of a trapezoid

(continued on the next page)

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Student-Built Glossary (continued)6

Vocabulary TermFound

on PageDefinition/Description/Example

parallelogram

rectangle

rhombus

square

trapezoid


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Anticipation GuideQuadrilaterals

Before you begin Chapter 6

• Read each statement.

• Decide whether you Agree (A) or Disagree (D) with the statement.

• Write A or D in the first column OR if you are not sure whether you agree or disagree, write NS (Not Sure).

After you complete Chapter 6

• Reread each statement and complete the last column by entering an A or a D.

• Did any of your opinions about the statements change from the first column?

• For those statements that you mark with a D, use a piece of paper to write an example of why you disagree.

6

STEP 1A, D, or NS

StatementSTEP 2A or D

1. A triangle has no diagonals.

2. A diagonal of a polygon is a segment joining the midpoints of two sides of the polygon.

3. The sum of the measures of the angles in a polygon can be determined by subtracting 2 from the number of sides and multiplying the result by 180.

4. For a quadrilateral to be a parallelogram it must have two pairs of parallel sides.

5. The diagonals of a parallelogram are congruent.

6. If you know that one pair of opposite sides of a quadrilateral is both parallel and congruent, then you know the quadrilateral is a parallelogram.

7. If a quadrilateral is a rectangle, then all four angles are congruent.

8. The diagonals of a rhombus are congruent.

9. The properties of a rhombus are not true for a square.

10. A trapezoid has only one pair of parallel sides.

11. The median of a trapezoid is perpendicular to the bases.

12. An isosceles trapezoid has exactly one pair of congruent sides.

Step 2

Step 1


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Antes de comenzar el Capítulo 6

• Lee cada enunciado.

• Decide si estás de acuerdo (A) o en desacuerdo (D) con el enunciado.

• Escribe A o D en la primera columna O si no estás seguro(a) de la respuesta, escribe NS (No estoy seguro(a).

Después de completar el Capítulo 6

• Vuelve a leer cada enunciado y completa la última columna con una A o una D.

• ¿Cambió cualquiera de tus opiniones sobre los enunciados de la primera columna?

• En una hoja de papel aparte, escribe un ejemplo de por qué estás en desacuerdo con los enunciados que marcaste con una D.

Ejercicios preparatoriosCuadriláteros

6

Paso 1

PASO 1A, D o NS

EnunciadoPASO 2A o D

1. Un triángulo no tiene diagonales.

2. La diagonal de un polígono es un segmento que une los puntos medios de dos lados del polígono.

3. La suma de las medidas de los ángulos de un polígono puede determinarse restando 2 del número de lados y multiplicando el resultado por 180.

4. Para que un cuadrilátero sea un paralelogramo debe tener dos pares de lados paralelos.

5. Las diagonales de un paralelogramo son congruentes.

6. Si sabes que un par de lados opuestos de un cuadrilátero son paralelos y congruentes, entonces sabes que el cuadrilátero es un paralelogramo.

7. Si un cuadrilátero es un rectángulo, entonces todos los cuatro lados son congruentes.

8. Las diagonales de un rombo son congruentes.

9. Las propiedades de un rombo no son verdaderas para un cuadrado.

10. Un trapecio sólo tiene dos lados paralelos.

11. La mediana de un trapecio es perpendicular a las bases.

12. Un trapecio isósceles tiene exactamente un par de lados congruentes.

Paso 2

Capítulo 6 4 Geometría de Glencoe


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Study Guide and InterventionAngles of Polygons

Polygon Interior Angles Sum The segments that connect the nonconsecutive vertices of a polygon are called diagonals. Drawing all of the diagonals from one vertex of an n-gon separates the polygon into n - 2 triangles. The sum of the measures of the interior angles of the polygon can be found by adding the measures of the interior angles of those n - 2 triangles.

A convex polygon has 13 sides. Find the sum of the measures of the interior angles.

(n - 2) � 180 = (13 - 2) � 180= 11 �

180= 1980

The measure of an interior angle of a regular polygon is 120. Find the number of sides.

The number of sides is n, so the sum of the measures of the interior angles is 120n. 120n = (n - 2) � 180 120n = 180n - 360-60n = -360 n = 6

ExercisesFind the sum of the measures of the interior angles of each convex polygon.

1. decagon 2. 16-gon 3. 30-gon

4. octagon 5. 12-gon 6. 35-gon

The measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon.

7. 150 8. 160 9. 175

10. 165 11. 144 12. 135

13. Find the value of x.

6 -1

7x°

(4x + 10)°

(4x + 5)°

(6x + 10)°

(5x - 5)°

A B

C

D

E

Polygon Interior Angle

Sum TheoremThe sum of the interior angle measures of an n-sided convex polygon is (n - 2) · 180.

Example 1 Example 2


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Study Guide and Intervention (continued)

Angles of Polygons

Polygon Exterior Angles Sum There is a simple relationship among the exterior angles of a convex polygon.

Find the sum of the measures of the exterior angles, one at each vertex, of a convex 27-gon.

For any convex polygon, the sum of the measures of its exterior angles, one at each vertex, is 360.

Find the measure of each exterior angle of

regular hexagon ABCDEF.

The sum of the measures of the exterior angles is 360 and a hexagon has 6 angles. If n is the measure of each exterior angle, then 6n = 360 n = 60The measure of each exterior angle of a regular hexagon is 60.

ExercisesFind the sum of the measures of the exterior angles of each convex polygon.

1. decagon 2. 16-gon 3. 36-gon

Find the measure of each exterior angle for each regular polygon.

4. 12-gon 5. hexagon 6. 20-gon

7. 40-gon 8. heptagon 9. 12-gon

10. 24-gon 11. dodecagon 12. octagon

A B

C

DE

F

6 -1

Polygon Exterior Angle

Sum Theorem

The sum of the exterior angle measures of a convex polygon, one angle

at each vertex, is 360.

Example 1

Example 2


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Skills PracticeAngles of Polygons

Find the sum of the measures of the interior angles of each convex polygon.

1. nonagon 2. heptagon 3. decagon


4. 108 5. 120 6. 150

Find the measure of each interior angle.

7. 8.

9. 10.

Find the measures of each interior angle of each regular polygon.

11. quadrilateral 12. pentagon 13. dodecagon

Find the measures of each exterior angle of each regular polygon.

14. octagon 15. nonagon 16. 12-gon

6-1

x°

x°

(2x - 15)°

(2x - 15)°

A B

CD(2x)°

(2x + 20)° (3x - 10)°

(2x - 10)°

L M

NP

(2x + 16)°

(x + 14)° (x + 14)°

(2x + 16)° (7x)° (7x)°

(4x)° (4x)°

(7x)° (7x)°


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Find the sum of the measures of the interior angles of each convex polygon.

1. 11-gon 2. 14-gon 3. 17-gon


4. 144 5. 156 6. 160

Find the measure of each interior angle.

7. 8.

Find the measures of an exterior angle and an interior angle given the number of sides of each regular polygon. Round to the nearest tenth, if necessary.

9. 16 10. 24 11. 30

12. 14 13. 22 14. 40

15. CRYSTALLOGRAPHY Crystals are classified according to seven crystal systems. The basis of the classification is the shapes of the faces of the crystal. Turquoise belongs to the triclinic system. Each of the six faces of turquoise is in the shape of a parallelogram. Find the sum of the measures of the interior angles of one such face.

PracticeAngles of Polygons

6 -1

x°

(2x + 15)° (3x - 20)°

(x + 15)°

J K

MN

(6x - 4)°

(2x + 8)° (6x - 4)°

(2x + 8)°


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Word Problem PracticeAngles of Polygons

La Tribuna

1. ARCHITECTURE In the Uffizi gallery in Florence, Italy, there is a room built by Buontalenti called the Tribune (La Tribuna in Italian). This room is shaped like a regular octagon.

What angle do consecutive walls of the Tribune make with each other?

2. BOXES Jasmine is designing boxes she will use to ship her jewelry. She wants to shape the box like a regular polygon. In order for the boxes to pack tightly, she decides to use a regular polygon that has the property that the measure of its interior angles is half the measure of its exterior angles. What regular polygon should she use?

3. THEATER A theater floor plan is shown in the figure. The upper five sides are part of a regular dodecagon.

Find m∠1.

4. ARCHEOLOGY Archeologists unearthed parts of two adjacent walls of an ancient castle.

Before it was unearthed, they knew from ancient texts that the castle was shaped like a regular polygon, but nobody knew how many sides it had. Some said 6, others 8, and some even said 100. From the information in the figure, how many sides did the castle really have?

5. POLYGON PATH In Ms. Rickets’ math class, students made a “polygon path” that consists of regular polygons of 3, 4, 5, and 6 sides joined together as shown.

a. Find m∠2 and m∠5.

b. Find m∠3 and m∠4.

c. What is m∠1?

6 -1

2

1

34

5

stage1

24˚


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Enrichment

Central Angles of Regular PolygonsYou have learned about the interior and exterior angles of a polygon. Regular polygons also have central angles. A central angle is measured from the center of the polygon.

The center of a polygon is the point equidistant from all of the vertices of the polygon, just as the center of a circle is the point equidistant from all of the points on the circle. The central angle is the angle drawn with the vertex at the center of the circle and the sides of angle drawn through consecutive vertices of the polygon. One of the central angles that can be drawn in this regular hexagon is ∠APB. You may remember from making circle graphs that there are 360° around the center of a circle.

1. By using logic or by drawing sketches, find the measure of the central angle of each regular polygon.

2. Make a conjecture about how the measure of a central angle of a regular polygon relates to the measures of the interior angles and exterior angles of a regular polygon.

3. CHALLENGE In obtuse �ABC, −−−

BC is the longest side. −−

AC is also a side of a 21-sided regular polygon.

−−

AB is also a side of a 28-sided regular polygon. The 21-sided regular polygon and the 28-sided regular polygon have the same center point P. Find n if

−−−

BC is a side of a n-sided regular polygon that has center point P.

(Hint: Sketch a circle with center P and place points A, B, and C on the circle.)

6 -1

A

B

P

r


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Study Guide and InterventionParallelograms

Sides and Angles of Parallelograms A quadrilateral with both pairs of opposite sides parallel is a parallelogram. Here are four important properties of parallelograms.

If ABCD is a parallelogram, find the value of each variable.−−

AB and −−−

CD are opposite sides, so −−

AB � −−−

CD . 2a = 34 a = 17∠A and ∠C are opposite angles, so ∠A � ∠C. 8b = 112 b = 14

ExercisesFind the value of each variable.

1. 2.

3. 4.

5. 6.

P Q

RS

BA

D C34

2a8b°

112°

4y°

3x°8y

88

6x°

72x

30x 150

2y

6 -2

If PQRS is a parallelogram, then

If a quadrilateral is a parallelogram,

then its opposite sides are congruent.

−−−

PQ � −−

SR and −−

PS �

−−−

QR


then its opposite angles are congruent.

∠P � ∠R and ∠S � ∠Q


then its consecutive angles are

supplementary.

∠P and ∠S are supplementary; ∠S and ∠R are supplementary;

∠R and ∠Q are supplementary; ∠Q and ∠P are supplementary.

If a parallelogram has one right angle,

then it has four right angles.

If m∠P = 90, then m∠Q = 90, m∠R = 90, and m∠S = 90.

Example

3y

12

6x

60°

55° 5x°

2y°

6x° 12x°

3y°


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Parallelograms

Diagonals of Parallelograms Two important properties of parallelograms deal with their diagonals.

Find the value of x and y in parallelogram ABCD.

The diagonals bisect each other, so AE = CE and DE = BE. 6x = 24 4y = 18 x = 4 y = 4.5

ExercisesFind the value of each variable.

1. 2. 3.

4. 5. 6.

COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of �ABCD with the given vertices.

7. A(3, 6), B(5, 8), C(3, −2), and D(1, −4) 8. A(−4, 3), B(2, 3), C(−1, −2), and D(−7, −2)

9. PROOF Write a paragraph proof of the following.

Given: �ABCD Prove: �AED � �BEC

A B

CD

P

A BE

CD

6x

1824

4y

3x4y

812

4x

2y28

3x

30°10

y

A B

CDE

2x°

60° 4y°

3x°2y

12

6 -2

If ABCD is a parallelogram, then

If a quadrilateral is a parallelogram, then

its diagonals bisect each other.AP = PC and DP = PB

If a quadrilateral is a parallelogram, then each

diagonal separates the parallelogram

into two congruent triangles.

�ACD � �CAB and �ADB � �CBD

Example

x

17

4y

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Skills PracticeParallelograms

ALGEBRA Find the value of each variable.

1. 2.

3. 4.

5. 6.

COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of �HJKL with the given vertices.

7. H(1, 1), J(2, 3), K(6, 3), L(5, 1) 8. H(-1, 4), J(3, 3), K(3, -2), L(-1, -1)

9. PROOF Write a paragraph proof of the theorem Consecutive angles in a parallelogram are supplementary.

6-2

x - 2 y + 1

1926

x° 44°

y°

4x-

2 3y -1

y +5 x

+10

2b -1 b + 3

3a - 5

2a

a + 146a + 4

10b + 1

9b + 8

(x + 8)°

(3x)°

(y + 9)°


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PracticeParallelograms

ALGEBRA Find the value of each variable.

1. 2.

3. 4.

ALGEBRA Use �RSTU to find each measure or value.

5. m∠RST = 6. m∠STU =

7. m∠TUR = 8. b =

COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of �PRYZ with the given vertices.

9. P(2, 5), R(3, 3), Y(-2, -3), Z(-3, -1) 10. P(2, 3), R(1, -2), Y(-5, -7), Z(-4, -2)

11. PROOF Write a paragraph proof of the following. Given: �PRST and �PQVU

Prove: ∠V � ∠S

12. CONSTRUCTION Mr. Rodriquez used the parallelogram at the right to design a herringbone pattern for a paving stone. He will use the paving stone for a sidewalk. If m∠1 is 130, find m∠2, m∠3, and m∠4.

TU

SRB30° 23

4b - 1

25°

P R

S

U V

Q

T

1 2

34

6 -2

(4x)°

(y+10)°

(2y-40)°

4x

5y - 83y + 1

x +6

y+

3 15

x -3

12

b+1

2b

a+2

3a-4


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1. WALKWAY A walkway is made by adjoining four parallelograms as shown.

Are the end segments a and e parallel to each other? Explain.

2. DISTANCE Four friends live at the four corners of a block shaped like a parallelogram. Gracie lives 3 miles away from Kenny. How far apart do Teresa and Travis live from each other?

3. SOCCER Four soccer players are located at the corners of a parallelogram. Two of the players in opposite corners are the goalies. In order for goalie A to be able to see the three others, she must be able to see a certain angle x in her field of vision.

What angle does the other goalie have to be able to see in order to keep an eye on the other three players?

4. VENN DIAGRAMS Make a Venn diagram showing the relationship between squares, rectangles, and parallelograms.

5. SKYSCRAPERS On vacation, Tony’s family took a helicopter tour of the city. The pilot said the newest building in the city was the building with this top view. He told Tony that the exterior angle by the front entrance is 72°. Tony wanted to know more about the building, so he drew this diagram and used his geometry skills to learn a few more things. The front entrance is next to vertex B.

a. What are the measures of the four angles of the parallelogram?

b. How many pairs of congruent triangles are there in the figure? What are they?

Word Problem PracticeParallelograms

A B

C D

E

72˚

Goalie B Player A

Goalie A

xPlayer B

TeresaTravis

Gracie Kenny

a

e

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Enrichment

Diagonals of ParallelogramsIn some drawings the diagonal of a parallelogram appears to be the angle bisector of both opposite angles. When might that be true?

1. Given: Parallelogram PQRS with diagonal −−

PR . −−

PR is an angle bisector of ∠QPS and ∠QRS.

What type of parallelogram is PQRS? Justify your answer.

2. Given: Parallelogram WPRK with angle bisector

−−−

KD , DP = 5, and WD = 7.

Find WK and KR.

3. Refer to Exercise 2. Write a statement about parallelogram WPRK and angle bisector

−−−

KD .

4. Given: Parallelogram ABCD with diagonal

−−−

BD and angle bisector −−

BP . PD = 5, BP = 6, and CP = 6. The perimeter of triangle PCD is 15.

Find AB and BC.

P

Q R

S

PDA

BC

PD

RK

W

6 -2


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Study Guide and InterventionTests for Parallelograms

Conditions for Parallelograms There are many ways to establish that a quadrilateral is a parallelogram.

Find x and y so that FGHJ is a parallelogram.

FGHJ is a parallelogram if the lengths of the opposite sides are equal. 6x + 3 = 15 4x - 2y = 2 6x = 12 4(2) - 2y = 2 x = 2 8 - 2y = 2 -2y = -6 y = 3

ExercisesFind x and y so that the quadrilateral is a parallelogram.

1. 2.

3. 4.

5. 6.

A B

CD

E

J H

GF 6x + 3

154x - 2y 2

2x - 2

2y 8

12 11x°

5y° 55°

5x°5y°

25°

(x + y)°

2x°

30°

24°

6y°

3x°

9x°

6y45°

18

6 -3

If: If:

both pairs of opposite sides are parallel, −−

AB || −−−

DC and −−

AD || −−−

BC ,

both pairs of opposite sides are congruent, −−

AB �

−−−

DC and

−−

AD �

−−−

BC ,

both pairs of opposite angles are congruent, ∠ABC � ∠ADC and ∠DAB � ∠BCD,

the diagonals bisect each other, −−

AE � −−

CE and −−

DE � −−

BE ,

one pair of opposite sides is congruent and parallel, −−

AB || −−−

CD and −−

AB � −−−

CD , or −−

AD || −−−

BC and −−

AD � −−−

BC ,

then: the figure is a parallelogram. then: ABCD is a parallelogram.

Example


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Tests for Parallelograms

Parallelograms on the Coordinate Plane On the coordinate plane, the Distance, Slope, and Midpoint Formulas can be used to test if a quadrilateral is a parallelogram.

Determine whether ABCD is a parallelogram.

The vertices are A(-2, 3), B(3, 2), C(2, -1), and D(-3, 0).

Method 1: Use the Slope Formula, m = y2 - y1 − x2 - x1

.

slope of −−−

AD = 3 - 0 −

-2 - (-3) = 3 −

1 = 3 slope of

−−−

BC = 2 - (-1) −

3 - 2 = 3 −

1 = 3

slope of −−

AB =

2 - 3 −

3 - (-2) = -

1 −

5 slope of

−−−

CD = -1 - 0 −

2 - (-3) = -

1 −

5

Since opposite sides have the same slope, −−

AB || −−−

CD and −−−

AD || −−−

BC . Therefore, ABCD is a parallelogram by definition.

Method 2: Use the Distance Formula, d = √ �� (x2 - x1)2 + ( y2 - y1)

2 .

AB = √ �� (-2 - 3)2 + (3 - 2)2 = √ �� 25 + 1 or √ �� 26

CD = √ �� (2 - (-3))2 + (-1 - 0)2 = √ �� 25 + 1 or √ �� 26

AD = √ �� (-2 - (-3))2 + (3 - 0)2 = √ �� 1 + 9 or √ �� 10

BC = √ �� (3 - 2)2 + (2 - (-1))2 = √ �� 1 + 9 or √ �� 10

Since both pairs of opposite sides have the same length, −−

AB � −−−

CD and −−−

AD � −−−

BC . Therefore, ABCD is a parallelogram by Theorem 6.9.

ExercisesGraph each quadrilateral with the given vertices. Determine whether the figure is a parallelogram. Justify your answer with the method indicated.

1. A(0, 0), B(1, 3), C(5, 3), D(4, 0); 2. D(-1, 1), E(2, 4), F(6, 4), G(3, 1);Slope Formula Slope Formula

3. R(-1, 0), S(3, 0), T(2, -3), U(-3, -2); 4. A(-3, 2), B(-1, 4), C(2, 1), D(0, -1);Distance Formula Distance and Slope Formulas

5. S(-2, 4), T(-1, -1), U(3, -4), V(2, 1); 6. F(3, 3), G(1, 2), H(-3, 1), I(-1, 4);Distance and Slope Formulas Midpoint Formula

7. A parallelogram has vertices R(-2, -1), S(2, 1), and T(0, -3). Find all possible coordinates for the fourth vertex.

x

y

O

C

A

D

B

6 -3

Example


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Skills PracticeTests for Parallelograms

Determine whether each quadrilateral is a parallelogram. Justify your answer.

1. 2.

3. 4.

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a parallelogram. Justify your answer with the method indicated.

5. P(0, 0), Q(3, 4), S(7, 4), Y(4, 0); Slope Formula

6. S(-2, 1), R(1, 3), T(2, 0), Z(-1, -2); Distance and Slope Formulas

7. W(2, 5), R(3, 3), Y(-2, -3), N(-3, 1); Midpoint Formula

ALGEBRA Find x and y so that each quadrilateral is a parallelogram.

8. 9.

10. 11.

x + 16

y + 192y

2x - 82y - 11

y + 3 4x - 3

3x

(4x - 35)°

(3x + 10)°(2y - 5)°

(y + 15)°

3x - 14

x + 20

3y + 2y + 20

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Determine whether each quadrilateral is a parallelogram. Justify your answer.

1. 2.

3. 4.

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a parallelogram. Justify your answer with the method indicated.

5. P(-5, 1), S(-2, 2), F(-1, -3), T(2, -2); Slope Formula

6. R(-2, 5), O(1, 3), M(-3, -4), Y(-6, -2); Distance and Slope Formulas

ALGEBRA Find x and y so that the quadrilateral is a parallelogram.

7. 8.

9. 10.

11. TILE DESIGN The pattern shown in the figure is to consist of congruent parallelograms. How can the designer be certain that the shapes are parallelograms?

118° 62°

118°62°

(5x + 29)°

(7x - 11)°(3y + 15)°

(5y - 9)°

3y - 5-3x + 4

-4x - 22y + 8

-4x + 6

-6x

7y + 312y - 7

-4y - 2

x + 12

-2x + 6

y + 23

PracticeTests for Parallelograms

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1. BALANCING Nikia, Madison, Angela, and Shelby are balancing themselves on an “X”-shaped floating object. To balance themselves, they want to make themselves the vertices of a parallelogram.

In order to achieve this, do all four of them have to be the same distance from the center of the object? Explain.

2. COMPASSES Two compass needles placed side by side on a table are both 2 inches long and point due north. Do they form the sides of a parallelogram?

3. FORMATION Four jets are flying in formation. Three of the jets are shown in the graph. If the four jets are located at the vertices of a parallelogram, what are the three possible locations of the missing jet?

4. STREET LAMPS When a coordinate plane is placed over the Harrisville town map, the four street lamps in the center are located as shown. Do the four lamps form the vertices of a parallelogram? Explain.

5. PICTURE FRAME Aaron is making a wooden picture frame in the shape of a parallelogram. He has two pieces of wood that are 3 feet long and two that are 4 feet long.

a. If he connects the pieces of wood at their ends to each other, in what order must he connect them to make a parallelogram?

b. How many different parallelograms could he make with these four lengths of wood?

c. Explain something Aaron might do to specify precisely the shape of the parallelogram.

5

5 y

xO–5

–5

5

5

y

xO

Shelby

Madison

NikiaAngela

6 -3 Word Problem PracticeTests for Parallelograms


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Enrichment

Tests for Parallelograms

By definition, a quadrilateral is a parallelogram if and only if both pairs of opposite sides are parallel. What conditions other than both pairs of opposite sides parallel will guarantee that a quadrilateral is a parallelogram? In this activity, several possibilities will be investigated by drawing quadrilaterals to satisfy certain conditions. Remember that any test that seems to work is not guaranteed to work unless it can be formally proven.

Complete.

1. Draw a quadrilateral with one pair of opposite sides congruent. Must it be a parallelogram?

2. Draw a quadrilateral with both pairs of opposite sides congruent. Must it be a parallelogram?

3. Draw a quadrilateral with one pair of opposite sides parallel and the other pair of opposite sides congruent. Must it be a parallelogram?

4. Draw a quadrilateral with one pair of opposite sides both parallel and congruent. Must it be a parallelogram?

5. Draw a quadrilateral with one pair of opposite angles congruent. Must it be a parallelogram?

6. Draw a quadrilateral with both pairs of opposite angles congruent. Must it be a parallelogram?

7. Draw a quadrilateral with one pair of opposite sides parallel and one pair of opposite angles congruent. Must it be a parallelogram?

6 -3


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Study Guide and InterventionRectangles

Properties of Rectangles A rectangle is a quadrilateral with four right angles. Here are the properties of rectangles.A rectangle has all the properties of a parallelogram.• Opposite sides are parallel.• Opposite angles are congruent.• Opposite sides are congruent.• Consecutive angles are supplementary.• The diagonals bisect each other.Also:• All four angles are right angles. ∠UTS, ∠TSR, ∠SRU, and ∠RUT are right angles.• The diagonals are congruent.

−−

TR � −−−

US

T S

RU

Q

Quadrilateral RUTS above is a rectangle. If US = 6x + 3 and RT = 7x - 2, find x.

The diagonals of a rectangle are congruent, so US = RT. 6x + 3 = 7x - 2 3 = x - 2 5 = x

Quadrilateral RUTS above is a rectangle. If m∠STR = 8x + 3 and m∠UTR = 16x - 9, find m∠STR.

∠UTS is a right angle, so m∠STR + m∠UTR = 90. 8x + 3 + 16x - 9 = 90 24x - 6 = 90 24x = 96 x = 4m∠STR = 8x + 3 = 8(4) + 3 or 35

ExercisesQuadrilateral ABCD is a rectangle.

1. If AE = 36 and CE = 2x - 4, find x.

2. If BE = 6y + 2 and CE = 4y + 6, find y.

3. If BC = 24 and AD = 5y - 1, find y.

4. If m∠BEA = 62, find m∠BAC.

5. If m∠AED = 12x and m∠BEC = 10x + 20, find m∠AED.

6. If BD = 8y - 4 and AC = 7y + 3, find BD.

7. If m∠DBC = 10x and m∠ACB = 4x2- 6, find m∠ ACB.

8. If AB = 6y and BC = 8y, find BD in terms of y.

B C

DA

E

6 -4

Example 1 Example 2


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Rectangles

Prove that Parallelograms Are Rectangles The diagonals of a rectangle are congruent, and the converse is also true.

If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

In the coordinate plane you can use the Distance Formula, the Slope Formula, and properties of diagonals to show that a figure is a rectangle.

Quadrilateral ABCD has vertices A(-3, 0), B(-2, 3), C(4, 1), and D(3, -2). Determine whether ABCD is a rectangle.

Method 1: Use the Slope Formula.

slope of −−

AB = 3 - 0 −

-2 - (-3) = 3 −

1 or 3 slope of

−−−

AD = -2 - 0 −

3 - (-3) = -2 −

6 or - 1 −

3

slope of −−−

CD = -2 - 1 −

3 - 4 = -3 −

-1 or 3 slope of

−−−

BC =

1 - 3 −

4 - (-2) = -2 −

6 or - 1 −

3

Opposite sides are parallel, so the figure is a parallelogram. Consecutive sides are perpendicular, so ABCD is a rectangle.

x

y

O

D

B

A

E C

6 -4

Example

Method 2: Use the Distance Formula.

AB = √ �� (-3 - (-2) ) 2 + (0 - 3 ) 2 or √ �� 10 BC = √ ��

(-2 - 4 ) 2 + (3 - 1 ) 2 or √ �� 40

CD = √ �� (4 - 3 ) 2 + (1 - (-2) ) 2 or √ �� 10 AD = √ ��

(-3 - 3 ) 2 + (0 - (-2) ) 2 or √ �� 40

Opposite sides are congruent, thus ABCD is a parallelogram.

AC = √ �� (-3 - 4 ) 2 + (0 - 1 ) 2 or √ �� 50 BD = √ ��

(-2 - 3 ) 2 + (3 - (-2) ) 2 or √ �� 50

ABCD is a parallelogram with congruent diagonals, so ABCD is a rectangle.

ExercisesCOORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula.

1. A(−3, 1), B(−3, 3), C(3, 3), D(3, 1); Distance Formula

2. A(−3, 0), B(−2, 3), C(4, 5), D(3, 2); Slope Formula

3. A(−3, 0), B(−2, 2), C(3, 0), D(2 −2); Distance Formula

4. A(−1, 0), B(0, 2), C(4, 0), D(3, −2); Distance Formula


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Skills PracticeRectangles

ALGEBRA Quadrilateral ABCD is a rectangle.

1. If AC = 2x + 13 and DB = 4x - 1, find DB.

2. If AC = x + 3 and DB = 3x - 19, find AC.

3. If AE = 3x + 3 and EC = 5x - 15, find AC.

4. If DE = 6x - 7 and AE = 4x + 9, find DB.

5. If m∠DAC = 2x + 4 and m∠BAC = 3x + 1, find m∠BAC.

6. If m∠BDC = 7x + 1 and m∠ADB = 9x - 7, find m∠BDC.

7. If m∠ABD = 7x - 31 and m∠CDB = 4x + 5, find m∠ABD.

8. If m∠BAC = x + 3 and m∠CAD = x + 15, find m∠BAC.

9. PROOF: Write a two-column proof.

Given: RSTV is a rectangle and U is the midpoint of

−−

VT . Prove: �RUV � �SUT

Statements Reasons

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula.

10. P(-3, -2), Q(-4, 2), R(2, 4), S(3, 0); Slope Formula

11. J(-6, 3), K(0, 6), L(2, 2), M(-4, -1); Distance Formula

12. T(4, 1), U(3, -1), X(-3, 2), Y(-2, 4); Distance Formula

D C

BAE

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PracticeRectangles

ALGEBRA Quadrilateral RSTU is a rectangle.

1. If UZ = x + 21 and ZS = 3x - 15, find US.

2. If RZ = 3x + 8 and ZS = 6x - 28, find UZ.

3. If RT = 5x + 8 and RZ = 4x + 1, find ZT.

4. If m∠SUT = 3x + 6 and m∠RUS = 5x - 4, find m∠SUT.

5. If m∠SRT = x + 9 and m∠UTR = 2x - 44, find m∠UTR.

6. If m∠RSU = x + 41 and m∠TUS = 3x + 9, find m∠RSU.

Quadrilateral GHJK is a rectangle. Find each measure if m∠1 = 37.

7. m∠2 8. m∠3

9. m∠4 10. m∠5

11. m∠6 12. m∠7

COORDINATE GEOMETRY Graph each quadrilateral with the given vertices. Determine whether the figure is a rectangle. Justify your answer using the indicated formula.

13. B(-4, 3), G(-2, 4), H(1, -2), L(-1, -3); Slope Formula

14. N(-4, 5), O(6, 0), P(3, -6), Q(-7, -1); Distance Formula

15. C(0, 5), D(4, 7), E(5, 4), F(1, 2); Slope Formula

16. LANDSCAPING Huntington Park officials approved a rectangular plot of land for a Japanese Zen garden. Is it sufficient to know that opposite sides of the garden plot are congruent and parallel to determine that the garden plot is rectangular? Explain.

U T

SRZ

K

2 1 5

436

7

J

HG

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1. FRAMES Jalen makes the rectangular frame shown.

In order to make sure that it is a rectangle, Jalen measures the distances BD and AC. How should these two distances compare if the frame is a rectangle?

2. BOOKSHELVES A bookshelf consists of two vertical planks with five horizontal shelves. Are each of the four sections for books rectangles? Explain.

3. LANDSCAPING A landscaper is marking off the corners of a rectangular plot of land. Three of the corners are in place as shown.

What are the coordinates of the fourth corner?

4. SWIMMING POOLS Antonio is designing a swimming pool on a coordinate grid. Is it a rectangle? Explain.

5. PATTERNS Veronica made the pattern shown out of 7 rectangles with four equal sides. The side length of each rectangle is written inside the rectangle.

a. How many rectangles can be formed using the lines in this figure?

b. If Veronica wanted to extend her pattern by adding another rectangle with 4 equal sides to make a larger rectangle, what are the possible side lengths of rectangles that she can add?

Word Problem PracticeRectangles

1 1

2 3

5

8

5

y

xO

5

5y

xO–5

–5

A B

D C

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Constant PerimeterDouglas wants to fence a rectangular region of his back yard for his dog. He bought 200 feet of fence.

1. Complete the table to show the dimensions of five different rectangular pens that would use the entire 200 feet of fence. Then find the area of each rectangular pen.

2. Do all five of the rectangular pens have the same area? If not, which one has the larger area?

3. Write a rule for finding the dimensions of a rectangle with the largest possible area for a given perimeter.

4. Let x represent the length of a rectangle and y the width. Write the formula for all rectangles with a perimeter of 200. Then graph this relationship on the coordinate plane at the right.

Julio read that a dog the size of his new pet, Bennie, should have at least 100 square feet in his pen. Before going to the store to buy fence, Julio made a table to determine the dimensions for Bennie’s rectangular pen.

5. Complete the table to find five possible dimensions of a rectangular fenced area of 100 square feet.

6. Julio wants to save money by purchasing the least number feet of fencing to enclose the 100 square feet. What will be the dimensions of the completed pen?

7. Write a rule for finding the dimensions of a rectangle with the least possible perimeter for a given area.

8. For length x and width y, write a formula for the area of a rectangle with an area of 100 square feet. Then graph the formula.

100

100

y

xO

y

xO 100

100

6-4

Perimeter Length Width Area

200 80

200 70

200 60

200 50

200 45

Area Length WidthHow much

fence to buy

100

100

100

100

100

Enrichment


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Graphing Calculator ActivityTI-Nspire: Exploring Rectangles

A quadrilateral with four right angles is a rectangle. The TI-Nspire can be used to explore some of the characteristics of a rectangle. Use the following steps to draw a rectangle.

Step 1 Set up the calculator in the correct mode.• Choose Graphs & Geometry from the Home Menu.• From the View menu, choose 4: Hide Axis

Step 2 Draw the rectangle• From the 8: Shapes menu choose 3: Rectangle.• Click once to define the corner of the rectangle. Then move and click again. The

side of the rectangle is now defined. Move perpendicularly to draw the rectangle. Click to anchor the shape.

Step 3 Measure the lengths of the sides of the rectangle. • From the 7: Measurement menu choose 1: Length (Note that when you scroll

over the rectangle, the value now shown is the perimeter of the rectangle.)• Select each endpoint of a segment of the rectangle. Then click or press Enter to

anchor the length of the segment in the work area.• Repeat for the other sides of the rectangle.

Exercises 1. What appears to be true about the opposite sides of the rectangle?

2. Draw the diagonals of the rectangle using 5: Segment from the 6: Points and Lines Menu. Click on two opposite vertices to draw the diagonal. Repeat to draw the other diagonal.a. Measure each diagonal using the measurement tool. What do you observe?b. What is true about the triangles formed by the sides of the rectangle and a diagonal?

Justify your conclusion.

3. Press Clear three times and select Yes to clear the screen. Repeat the steps and draw another rectangle. Do the relationships that you found for the first rectangle you drew hold true for this rectangle?

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Geometer’s Sketchpad ActivityExploring Rectangles

A quadrilateral with four right angles is a rectangle. The Geometer’s Sketchpad is a useful tool for exploring some of the characteristics of a rectangle. Use the following steps to draw a rectangle.

Step 1 Use the Line tool to draw a line anywhere on the screen.

Step 2 Use the Point tool to draw a point that is not on the line. To draw a line perpendicular to the first line you drew, select the first line and the point. Then choose Perpendicular Line from the Construct menu.

Step 3 Use the Point tool to draw a point that is not on either of the lines you have drawn. Repeat the procedure in Step 2 to draw lines perpendicular to the two lines you have drawn.

A rectangle is formed by the segments whose endpoints are the points of intersection of the lines.

ExercisesUse the measuring capabilities of The Geometer’s Sketchpad to explore the characteristics of a rectangle.

1. What appears to be true about the opposite sides of the rectangle that you drew? Make a conjecture and then measure each side to check your conjecture.

2. Draw the diagonals of the rectangle by using the Selection Arrow tool to choose two opposite vertices. Then choose Segment from the Construct menu to draw the diagonal. Repeat to draw the other diagonal.

a. Measure each diagonal. What do you observe?

b. What is true about the triangles formed by the sides of the rectangle and a diagonal? Justify your conclusion.

3. Choose New Sketch from the File menu and follow steps 1–3 to draw another rectangle. Do the relationships you found for the first rectangle you drew hold true for this rectangle also?

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Study Guide and InterventionRhombi and Squares

Properties of Rhombi and Squares A rhombus is a quadrilateral with four congruent sides. Opposite sides are congruent, so a rhombus is also a parallelogram and has all of the properties of a parallelogram. Rhombi also have the following properties.

A square is a parallelogram with four congruent sides and four congruent angles. A square is both a rectangle and a rhombus; therefore, all properties of parallelograms, rectangles, and rhombi apply to squares.

In rhombus ABCD, m∠BAC = 32. Find the measure of each numbered angle.

ABCD is a rhombus, so the diagonals are perpendicular and �ABE is a right triangle. Thus m∠4 = 90 and m∠1 = 90 - 32 or 58. The diagonals in a rhombus bisect the vertex angles, so m∠1 = m∠2. Thus, m∠2 = 58.A rhombus is a parallelogram, so the opposite sides are parallel. ∠BAC and ∠3 are alternate interior angles for parallel lines, so m∠3 = 32.

ExercisesQuadrilateral ABCD is a rhombus. Find each value or measure.

1. If m∠ABD = 60, find m∠BDC. 2. If AE = 8, find AC.

3. If AB = 26 and BD = 20, find AE. 4. Find m∠CEB.

5. If m∠CBD = 58, find m∠ACB. 6. If AE = 3x - 1 and AC = 16, find x.

7. If m∠CDB = 6y and m∠ACB = 2y + 10, find y.

8. If AD = 2x + 4 and CD = 4x - 4, find x.

M O

HRB

CA

D

E

B

32°

1 2

34

C

A D

EB

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The diagonals are perpendicular. −−−

MH ⊥

−−−

RO

Each diagonal bisects a pair of opposite angles. −−−

MH bisects ∠RMO and ∠RHO. −−−

RO bisects ∠MRH and ∠MOH.

Example


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Rhombi and Squares

Conditions for Rhombi and Squares The theorems below can help you prove that a parallelogram is a rectangle, rhombus, or square.

If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

If one diagonal of a parallelogram bisects a pair of opposite angles, then the parallelogram is a rhombus.

If one pair of consecutive sides of a parallelogram are congruent, the parallelogram is a rhombus.

If a quadrilateral is both a rectangle and a rhombus, then it is a square.

Determine whether parallelogram ABCD with vertices A(−3, −3), B(1, 1), C(5, −3), D(1, −7) is a rhombus, a rectangle, or a square.

AC = √ �� (-3 - 5)2 + ((-3 - (-3))2 = √ �� 64 = 8

BD = √ �� (1-1)2 + (-7 - 1)2 = √ �� 64 = 8The diagonals are the same length; the figure is a rectangle.

Slope of −−

AC =

-3 - (-3) −

-3 - 5 = 0 −

-8 = 8 The line is horizontal.

Slope of −−−

BD =

1 - (-7) − 1 - 1 = 8 −

0 = undefined The line is vertical.

Since a horizontal and vertical line are perpendicular, the diagonals are perpendicular. Parallelogram ABCD is a square which is also a rhombus and a rectangle.

ExercisesGiven each set of vertices, determine whether �ABCD is a rhombus, rectangle, or square. List all that apply. Explain.

1. A(0, 2), B(2, 4), C(4, 2), D(2, 0) 2. A(-2, 1), B(-1, 3), C(3, 1), D(2, -1)

3. A(-2, -1), B(0, 2), C(2, -1), D(0, -4) 4. A(-3, 0), B(-1, 3), C(5, -1), D(3, -4)

5. PROOF Write a two-column proof.Given: Parallelogram RSTU.

−−

RS � −−

ST Prove: RSTU is a rhombus.

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Skills PracticeRhombi and Squares

ALGEBRA Quadrilateral DKLM is a rhombus.

1. If DK = 8, find KL.

2. If m∠DML = 82 find m∠DKM.

3. If m∠KAL = 2x – 8, find x.

4. If DA = 4x and AL = 5x – 3, find DL.

5. If DA = 4x and AL = 5x – 3, find AD.

6. If DM = 5y + 2 and DK = 3y + 6, find KL.

7. PROOF Write a two-column proof.Given: RSTU is a parallelogram. −−−

RX �−−

TX �−−

SX �−−−

UX Prove: RSTU is a rectangle.

Statements Reasons

COORDINATE GEOMETRY Given each set of vertices, determine whether �QRSTis a rhombus, a rectangle, or a square. List all that apply. Explain.

8. Q(3, 5), R(3, 1), S(-1, 1), T(-1, 5)

9. Q(-5, 12), R(5, 12), S(-1, 4), T(-11, 4)

10. Q(-6, -1), R(4, -6), S(2, 5), T(-8, 10)

11. Q(2, -4), R(-6, -8), S(-10, 2), T(-2, 6)

M L

AD K

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PracticeRhombi and Squares

PRYZ is a rhombus. If RK = 5, RY = 13 and m∠YRZ = 67, find each measure.

1. KY

2. PK

3. m∠YKZ

4. m∠PZR

MNPQ is a rhombus. If PQ = 3 √ � 2 and AP = 3, find each measure.

5. AQ

6. m∠APQ

7. m∠MNP

8. PM

COORDINATE GEOMETRY Given each set of vertices, determine whether �BEFG is a rhombus, a rectangle, or a square. List all that apply. Explain.

9. B(-9, 1), E(2, 3), F(12, -2), G(1, -4)

10. B(1, 3), E(7, -3), F(1, -9), G(-5, -3)

11. B(-4, -5), E(1, -5), F(-2, -1), G(-7, -1)

12. TESSELLATIONS The figure is an example of a tessellation. Use a ruler or protractor to measure the shapes and then name the quadrilaterals used to form the figure.

K

P

YR

Z

AN

M Q

P

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1. TRAY RACKS A tray rack looks like a parallelogram from the side. The levels for the trays are evenly spaced.

What two labeled points form a rhombus with base AA' ?

2. SLICING Charles cuts a rhombus along both diagonals. He ends up with four congruent triangles. Classify these triangles as acute, obtuse, or right.

3. WINDOWS The edges of a window are drawn in the coordinate plane.

Determine whether the window is a square or a rhombus.

4. SQUARES Mackenzie cut a square along its diagonals to get four congruent right triangles. She then joined two of them along their long sides. Show that the resulting shape is a square.

5. DESIGN Tatianna made the design shown. She used 32 congruent rhombi to create the flower-like design at each corner.

a. What are the angles of the corner rhombi?

b. What kinds of quadrilaterals are the dotted and checkered figures?

Word Problem PracticeRhombi and Squares

y

xO

AB

CD

EF

G

H

B ,C ,D

,E,F ,G ,H ,

A20 inches

AB = 4 inches,

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Enrichment

Creating Pythagorean Puzzles

By drawing two squares and cutting them in a certain way, you can make a puzzle that demonstrates the Pythagorean Theorem. A sample puzzle is shown. You can create your own puzzle by following the instructions below.

1. Carefully construct a square and label the length of a side as a. Then construct a smaller square to the right of it and label the length of a side as b, as shown in the figure above. The bases should be adjacent and collinear.

2. Mark a point X that is b units from the left edge of the larger square. Then draw the segments from the upper left corner of the larger square to point X, and from point X to the upper right corner of the smaller square.

3. Cut out and rearrange your five pieces to form a larger square. Draw a diagram to show your answer.

4. Verify that the length of each side is equal to

√ �� a2 + b2 .

a

a

b X

b

b

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Study Guide and InterventionTrapezoids and Kites

Properties of Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. The midsegment or median of a trapezoid is the segment that connects the midpoints of the legs of the trapezoid. Its measure is equal to one-half the sum of the lengths of the bases. If the legs are congruent, the trapezoid is an isosceles trapezoid. In an isosceles trapezoid both pairs of base angles are congruent and the diagonals are congruent.

The vertices of ABCD are A(-3, -1), B(-1, 3), C(2, 3), and D(4, -1). Show that ABCD is a trapezoid and determine whether it is an isosceles trapezoid.

slope of −−

AB = 3 - (-1) −

-1 - (-3) = 4 −

2 = 2

slope of −−−

AD = -1 - (-1) −

4 - (-3) = 0 −

7 = 0

slope of −−−

BC = 3 - 3 −

2 - (-1) = 0 −

3 = 0

slope of −−−

CD = -1 - 3 −

4 - 2 = -4 −

2 = -2

Exactly two sides are parallel, −−−

AD and −−−

BC , so ABCD is a trapezoid. AB = CD, so ABCD is an isosceles trapezoid.

ExercisesFind each measure.

1. m∠D 2. m∠L

125°

5

5

40°

COORDINATE GEOMETRY For each quadrilateral with the given vertices, verify that the quadrilateral is a trapezoid and determine whether the figure is an isosceles trapezoid.

3. A(−1, 1), B(3, 2), C(1,−2), D(−2, −1) 4. J(1, 3), K(3, 1), L(3, −2), M(−2, 3)

For trapezoid HJKL, M and N are the midpoints of the legs.

5. If HJ = 32 and LK = 60, find MN.

6. If HJ = 18 and MN = 28, find LK.

T

R U

base

base

Sleg

x

y

O

B

DA

C

6-6

STUR is an isosceles trapezoid.

−−

SR � −−

TU ; ∠R � ∠U, ∠S � ∠T

Example

AB = √ �� (-3 - (-1))2 + (-1 - 3)2

= √ �� 4 + 16 = √ �� 20 = 2 √ � 5

CD = √ �� (2 - 4)2 + (3 - (-1))2

= √ �� 4 + 16 = √ �� 20 = 2 √ � 5

L K

NM

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Properties of Kites A kite is a quadrilateral with exactly two pairs of consecutive congruent sides. Unlike a parallelogram, the opposite sides of a kite are not congruent or parallel.

The diagonals of a kite are perpendicular.For kite RMNP,

−−−

MP ⊥ −−−

RN

In a kite, exactly one pair of opposite angles is congruent.For kite RMNP, ∠M � ∠P

If WXYZ is a kite, find m∠Z.

The measures of ∠Y and ∠W are not congruent, so ∠X � ∠Z.m∠X + m∠Y + m∠Z + m∠W = 360 m∠X + 60 + m∠Z + 80 = 360m∠X + m∠Z = 220m∠X = 110, m∠Z = 110

If ABCD is a kite, find BC.

The diagonals of a kite are perpendicular. Use the Pythagorean Theorem to find the missing length.BP2 + PC2 = BC2

52 + 122 = BC2

169 = BC2

13 = BC

ExercisesIf GHJK is a kite, find each measure.

1. Find m∠JRK.

2. If RJ = 3 and RK = 10, find JK.

3. If m∠GHJ = 90 and m∠GKJ = 110, find m∠HGK.

4. If HJ = 7, find HG.

5. If HG = 7 and GR = 5, find HR.

6. If m∠GHJ = 52 and m∠GKJ = 95, find m∠HGK.


Trapezoids and Kites

6-6

Example 1

Example 2

80°

60°

5

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ALGEBRA Find each measure. 1. m∠S 2. m∠M

63°

14 14142°

21 21

3. m∠D 4. RH

36° 70°20

12

12

ALGEBRA For trapezoid HJKL, T and S are midpoints of the legs.

5. If HJ = 14 and LK = 42, find TS.

6. If LK = 19 and TS = 15, find HJ.

7. If HJ = 7 and TS = 10, find LK.

8. If KL = 17 and JH = 9, find ST.

COORDINATE GEOMETRY EFGH is a quadrilateral with vertices E(1, 3), F(5, 0), G(8, −5), H(−4, 4).

9. Verify that EFGH is a trapezoid.

10. Determine whether EFGH is an isosceles trapezoid. Explain.

6-6 Skills PracticeTrapezoids and Kites


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PracticeTrapezoids and Kites

Find each measure.

1. m∠T 2. m∠Y

60°

68°

7

7

3. m∠Q 4. BC

110° 48°

11 47

7

ALGEBRA For trapezoid FEDC, V and Y are midpoints of the legs.

5. If FE = 18 and VY = 28, find CD.

6. If m∠F = 140 and m∠E = 125, find m∠D.

COORDINATE GEOMETRY RSTU is a quadrilateral with vertices R(−3, −3), S(5, 1), T(10, −2), U(−4, −9).

7. Verify that RSTU is a trapezoid.

8. Determine whether RSTU is an isosceles trapezoid. Explain.

9. CONSTRUCTION A set of stairs leading to the entrance of a building is designed in the shape of an isosceles trapezoid with the longer base at the bottom of the stairs and the shorter base at the top. If the bottom of the stairs is 21 feet wide and the top is 14 feet wide, find the width of the stairs halfway to the top.

10. DESK TOPS A carpenter needs to replace several trapezoid-shaped desktops in a classroom. The carpenter knows the lengths of both bases of the desktop. What other measurements, if any, does the carpenter need?

6-6

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Word Problem PracticeTrapezoids and Kites

1. PERSPECTIVE Artists use different techniques to make things appear to be 3-dimensional when drawing in two dimensions. Kevin drew the walls of a room. In real life, all of the walls are rectangles. In what shape did he draw the side walls to make them appear 3-dimensional?

2. PLAZA In order to give the feeling of spaciousness, an architect decides to make a plaza in the shape of a kite. Three of the four corners of the plaza are shown on the coordinate plane. If the fourth corner is in the first quadrant, what are its coordinates?

y

x

3. AIRPORTS A simplified drawing of the reef runway complex at Honolulu International Airport is shown below.

How many trapezoids are there in this image?

4. LIGHTING A light outside a room shines through the door and illuminates a trapezoidal region ABCD on the floor.

A B

D C

Under what circumstances would trapezoid ABCD be isosceles?

5. RISERS A riser is designed to elevate a speaker. The riser consists of 4 trapezoidal sections that can be stacked one on top of the other to produce trapezoids of varying heights.

20 feet

10 feet

All of the stages have the same height. If all four stages are used, the width of the top of the riser is 10 feet.

a. If only the bottom two stages are used, what is the width of the top of the resulting riser?

b. What would be the width of the riser if the bottom three stages are used?

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Enrichment

Quadrilaterals in Construction

Quadrilaterals are often used in construction work.

1. The diagram at the right represents a roof frame and shows many quadrilaterals. Find the following shapes in the diagram and shade in their edges.

a. isosceles triangle

b. scalene triangle

c. rectangle

d. rhombus

e. trapezoid (not isosceles)

f. isosceles trapezoid

2. The figure at the right represents a window. The wooden part between the panes of glass is 3 inches wide. The frame around the outer edge is 9 inches wide. The outside measurements of the frame are 60 inches by 81 inches. The height of the top and bottom panes is the same. The top three panes are the same size.

a. How wide is the bottom pane of glass?

b. How wide is each top pane of glass?

c. How high is each pane of glass?

3. Each edge of this box has been reinforced with a piece of tape. The box is 10 inches high, 20 inches wide, and 12 inches deep. What is the length of the tape that has been used?

jack rafters

hip rafter

common rafters

Roof Frame

ridge boardvalley rafter

plate

60 in.

81 in.

9 in.

3 in.

3 in.

9 in.

9 in.

10 in.

12 in.

20 in.

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SCORE 6

Read each question. Then fill in the correct answer.

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

Multiple Choice

Student Recording SheetUse this recording sheet with pages 452–453 of the Student Edition.

Record your answer in the blank.

For gridded response questions, also enter your answer in the grid by writing each number or symbol in a box. Then fill in the corresponding circle for that number or symbol.

8. (grid in)

9.

10.

11.

12. (grid in)

13. 9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

Extended Response

Short Response/Gridded Response

Record your answers for Question 14 on the back of this paper.

8. 12.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0


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6

General Scoring Guidelines

• If a student gives only a correct numerical answer to a problem but does not show how he or she arrived at the answer, the student will be awarded only 1 credit. All extended-response questions require the student to show work.

• A fully correct answer for a multiple-part question requires correct responses for all parts of the question. For example, if a question has three parts, the correct response to one or two parts of the question that required work to be shown is not considered a fully correct response.

• Students who use trial and error to solve a problem must show their method. Merely showing that the answer checks or is correct is not considered a complete response for full credit.

Exercise 14 Rubric

Score Specifi c Criteria

4

Students correctly determine the quadrilateral in part a is a parallelogram since opposite sides are congruent. Students correctly determine the quadrilateral in part b does not contain sufficient information to prove it is a parallelogram. The two horizontal sides must also be congruent. Students correctly determine that the quadrilateral in part c is a parallelogram since both pairs of opposite angles are congruent.

3 A generally correct solution, but may contain minor flaws in reasoning or computation.

2 A partially correct interpretation and/or solution to the problem.

1 A correct solution with no evidence or explanation.

0An incorrect solution indicating no mathematical understanding of the concept or task, or no solution is given.

Rubric for Scoring Extended Response


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SCORE

1. Find the sum of the measures of the interior angles of a convex 70-gon.

2. The measure of each interior angle of a regular polygon is 172. Find the number of sides in the polygon.

3. The measure of each exterior angle of a regular polygon is 18. Find the number of sides in the polygon.

4. Given parallelogram ABCD with C(5, 4), find the coordinates of A if the diagonals

−−

AC and −−−

BD intersect at (2, 7).

5. MULTIPLE CHOICE Find m∠1 in parallelogram ABCD.

A 64 C 46 B 58 D 36

1.

2.

3.

4.

5.

1. Determine whether this quadrilateral is a parallelogram. Justify your answer.

For Questions 2–4, write true or false.

2. A quadrilateral with two pairs of parallel sides is always a parallelogram.

3. The diagonals of a parallelogram are always perpendicular.

4. The slope of −−

AB and −−−

CD is 3 −

5 and the slope of

−−−

BC and −−−

AD is

- 5 −

3 . ABCD is a parallelogram.

5. Refer to parallelogram ABCD. If AB = 8 cm, what is the perimeter of the parallelogram?

1.

2.

3.

4.

5.

Chapter 6 Quiz 1(Lessons 6-1 and 6-2)

Chapter 6 Quiz 2(Lesson 6-3)

6

6

A

D

B

C

6 cm


110°

46° 1

D C

BA

SCORE


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SCORE

1. MULTIPLE CHOICE RSTV is a rhombus. Which of the following statements is NOT true?

A −−

RV � −−

TS B

−−

RV ⊥ −−

TS C

−−

RS ‖ −−

TV D ∠R � ∠T

For Questions 2 and 3, refer to trapezoid MNPQ.

2. Find m∠M.

3. Find m∠Q.

4. True or false. A quadrilateral that is a rectangle and a rhombus is a square.

5. �ABCD has vertices A(4, 0), B(0, 4), C(−4, 0), and D(0, −4). Determine whether ABCD is a rectangle, rhombus, or square. List all that apply.

1.

2.

3.

4.

5.

For Questions 1 and 2, refer to kite DEFC.

1. If m∠DCF = 34 and m∠DEF = 90, find m∠CDE.

2. If DR = 5 and RE = 5, find FE.

For Questions 3 and 4, refer to trapezoid NPQM where X and Y are midpoints of the sides.

3. If MQ = 15 and XY = 10, find NP.

4. If NP = 13 and MQ = 18, find XY.

5. If CDEF is a trapezoid with vertices C(0, 2), D(2, 4), E(7, 3), and F(1, −3), how can you prove that it is an isosceles trapezoid?

1.

2.

3.

4.

5.

6 Chapter 6 Quiz 3(Lessons 6-4 and 6-5)

Chapter 6 Quiz 4(Lesson 6-6)

(2x + 3)°

(x + 9)°

6 SCORE


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6 SCORE

1. Find the measure of each exterior angle of a regular 56-gon. Round to the nearest tenth.

A 3.2 B 6.4 C 173.6 D 9720

2. Given BE = 2x + 6 and ED = 5x - 12 in parallelogram ABCD, find BD.

F 6 H 18 G 12 J 36

3. If the slope of −−−

PQ is 2 −

3 and the slope of

−−−

QR is - 1 −

2 , find the slope of

−− SR so that

PQRS is a parallelogram.

A 2 −

3 B 3 −

2 C -

1 −

2 D 2

4. Find m∠W in parallelogram RSTW. F 17 H 55 G 33 J 125

5. Find the sum of the measures of the interior angles of a convex 48-gon. A 172.5 B 360 C 8280 D 8640

D

ECB

A

6.

7.

8.

9.

10.

1.

2.

3.

4.

5.

Part II

6. Find x.

7. ABCD is a parallelogram with m∠A = 138. Find m∠B.

8. Determine whether ABCD is a parallelogram if AB = 6, BC = 12, CD = 6, and DA = 12. Justify your answer.

9. In parallelogram MLKJ, find m∠MLK and m∠LKJ.

10. XYWZ is a quadrilateral with vertices W(1, −4), X(-4, 2), Y(1, −1), and Z(−2, −3). Determine if the quadrilateral is a parallelogram. Use slope to justify your answer.

Part I Write the letter for the correct answer in the blank at the right of each question.

Chapter 6 Mid-Chapter Test(Lessons 6-1 through 6-3)

(3x + 2)°(2x - 12)°

(x - 10)°(x + 30)°

(2x - 1)°

(x + 16)°

18°

12°


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6 SCORE

Choose from the terms above to complete each sentence.

1. A quadrilateral with only one pair of opposite sides parallel and the other pair of opposite sides congruent is a(n) .

2. A quadrilateral with two pairs of opposite sides parallel is a(n) .

3. A quadrilateral with only one pair of opposite sides parallel is a(n) .

4. A quadrilateral that is both a rectangle and a rhombus is a(n) .

5. A quadrilateral with four congruent sides is a(n) .

Write whether each sentence is true or false. If false, replace the underlined word or number to make a true sentence.

6. A quadrilateral with four right angles is a .

7. A quadrilateral with two pairs of congruent consecutive sides is a .

Choose the correct term to complete each sentence.

8. Segments that join opposite vertices in a quadrilateral are called (medians, diagonals).

9. The segment joining the midpoints of the nonparallel sides of a trapezoid is called the (median, diagonal).

Define each term in your own words.

10. base angles of an isosceles trapezoid

11. legs of a trapezoid

? 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

base

base angle

diagonal

isosceles trapezoid

legs

midsegment of a trapezoid

parallelogram

rectangle

rhombus

square

trapezoid

Chapter 6 Vocabulary Test

?

?

?

?

trapezoid

kite


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6 SCORE

Write the letter for the correct answer in the blank at the right of each question.

1. Find the sum of the measures of the interior angles of a convex 30-gon. A 5400 B 5040 C 360 D 168

2. Find the sum of the measures of the exterior angles of a convex 21-gon. F 21 G 180 H 360 J 3420

3. If the measure of each interior angle of a regular polygon is 108, find the measure of each exterior angle.

A 18 B 72 C 90 D 108

4. For parallelogram ABCD, find the value of x. F 4 H 16 G 10.25 J 21.5

5. Which of the following is a property of a parallelogram? A The diagonals are congruent. C The diagonals are perpendicular. B The diagonals bisect the angles. D The diagonals bisect each other.

6. Find the values of x and y so that ABCD will be a parallelogram.F x = 6, y = 42

G x = 6, y = 22 H x = 20, y = 42 J x = 20, y = 22

7. Find the value of x so that this quadrilateral is a parallelogram.

A 44 C 90 B 46 D 134

8. Parallelogram ABCD has vertices A(0, 0), B(2, 4), and C(10, 4). Find the coordinates of D.

F D(8, 0) G D(10, 0) H D(0, 4) J D(10, 8)

9. Which of the following is a property of all rectangles? A four congruent sides C diagonals are perpendicular B diagonals bisect the angles D four right angles

10. ABCD is a rectangle with diagonals −−

AC and −−−

BD . If AC = 2x + 10 and BD = 56, find the value of x.

F 23 G 33 H 78 J 122

11. ABCD is a rectangle with B(-5, 0), C(7, 0) and D(7, 3). Find the coordinates of A.

A A(-5, 7) B A(3, 5) C A(-5, 3) D A(7, -3)

x° 134°

134° 46°

(4x)°

(y - 10)°32°

24°

5x - 12

3x + 20

A D

CB

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Chapter 6 Test, Form 1


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12. For rhombus ABCD, find m∠1. F 45 H 90 G 60 J 120

13. Find m∠PRS in square PQRS. A 30 C 60 B 45 D 90

14. Choose a pair of base angles of trapezoid ABCD. F ∠A, ∠C H ∠A, ∠D G ∠B, ∠D J ∠D, ∠C

15. In trapezoid DEFG, find m∠D. A 44 C 108 B 72 D 136

16. The hood of Olivia’s car is the shape of a trapezoid. The base bordering the windshield measures 30 inches and the base at the front of the car measures 24 inches. What is the width of the median of the hood?

F 25 in. G 27 in. H 28 in. J 29 in.

17. The length of one base of a trapezoid is 44, the median is 36, and the other base is 2x + 10. Find the value of x.

A 9 B 17 C 21 D 40

18. Given trapezoid ABCD with median −−

EF , which of the following is true?

F EF =

1 −

2 AD H AB = EF

G AE = FD J EF = BC + AD −

2

19. PQRS is a kite. Find m∠S.

A 100 C 200 B 160 D 360

20. JKLM is a kite, find JM. F √ �� 29 H √ �� 13

G √ �� 89 J 11

Bonus Find x and m∠WYZ in rhombus XYZW.

X Y

ZW

(3x + 20)°

(x 2)°

DA

B CFE

D G

E F136°

72°

P S

Q R

1

A D

CB

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 6 Test, Form 1 (continued)

D C

A B

124° 36°

8 2

5

5


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6 SCORE


1. Find the sum of the measures of the interior angles of a convex 45-gon. A 8100 B 7740 C 360 D 172

2. Find the value of x. F 30 H 102 G 66 J 138

3. Find the sum of the measures of the exterior angles of a convex 39-gon. A 39 B 90 C 180 D 360

4. Which of the following is a property of a parallelogram? F Each pair of opposite sides is congruent. G Only one pair of opposite angles is congruent. H Each pair of opposite angles is supplementary. J There are four right angles.

5. For parallelogram ABCD, find m∠1. A 60 C 36 B 54 D 18

6. ABCD is a parallelogram with diagonals intersecting at E. If AE = 3x + 12 and EC = 27, find the value of x.

F 5 G 17 H 27 J 47

7. Find the values of x and y so that this quadrilateral is a parallelogram.

A x = 13, y = 24 C x = 7, y = 24 B x = 13, y = 6 D x = 7, y = 6

8. Find the value of x so that this quadrilateral is a parallelogram.F 12 H 36

G 24 J 132

9. Parallelogram ABCD has vertices A(8, 2), B(6, -4), and C(-5, -4). Find the coordinates of D.

A D(-5, 2) B D(-3, 2) C D(-2, 2) D D(-4, 8)

10. ABCD is a rectangle. If AC = 5x + 2 and BD = x + 22, find the value of x. F 5 G 6 H 11 J 26

11. Which of the following is true for all rectangles? A The diagonals are perpendicular. B The diagonals bisect the angles. C The consecutive sides are congruent. D The consecutive sides are perpendicular.

(2x + 60)°

(4x - 12)°

(2x + 30)°

(5x - 9)°

(y - 12)°(1–2y)°

120°

36°

1

A D

CB

(2x + 10)°

(x + 40)°

(x - 20)°x°

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Chapter 6 Test, Form 2A


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6

12. ABCD is a rectangle with B(-4, 6), C(-4, 2), and D(10, 2). Find the coordinates of A.

F A(6, 4) G A(10, 4) H A(2, 6) J A(10, 6)

13. For rhombus GHJK, find m∠1. A 22 C 68 B 44 D 90

14. The diagonals of square ABCD intersect at E. If AE = 2x + 6 and BD = 6x - 10, find AC.

F 11 G 28 H 56 J 90

15. ABCD is an isosceles trapezoid with A(10, -1), B(8, 3), and C(-1, 3). Find the coordinates of D.

A D(-3, -1) B D(-10, -11) C D(-1, 8) D D(-3, 3)

16. For isosceles trapezoid MNOP, find m∠MNP. F 44 H 80 G 64 J 116

17. The length of one base of a trapezoid is 19 inches and the length of the median is 16 inches. Find the length of the other base.

A 35 in. B 19 in. C 17.5 in. D 13 in.

18. Judith built a fence to surround her property. On a coordinate plane, the four corners of the fence are located at (-16, 1), (-6, 5), (4, 1), and (-6, -3). Which of the following most accurately describes the shape of Judith’s fence?

F square H rhombus G rectangle J trapezoid

19. For kite PQRS, find m∠S

A 248 C 112 B 68 D 124

20. ABCD is a parallelogram with coordinates A(4, 2), B(4, -1), C(-2, -1), and D(-2, 2). To prove that ABCD is a rectangle, you would plot the parallelogram on a coordinate plane and then find which of the following?

F measures of the angles H slopes of the diagonals G lengths of the diagonals J midpoints of the diagonals

Bonus Find the possible value(s) of x in rectangle JKLM. J K

LM

x 2 + 8

3x + 36

P

ON

M 36°

44°

22°

1G H

JK

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

Chapter 6 Test, Form 2A (continued)

22°


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6 SCORE


1. Find the sum of the measures of the interior angles of a convex 50-gon. A 9000 B 8640 C 360 D 172.8

2. Find the value of x. F 16 H 50 G 34 J 70

3. Find the sum of the measures of the exterior angles of a convex 65-gon. A 5.54 B 90 C 180 D 360

4. Which of the following is a property of all parallelograms? F Each pair of opposite angles is congruent. G Only one pair of opposite sides is congruent. H Each pair of opposite angles is supplementary. J There are four right angles.

5. For parallelogram ABCD, find m∠1. A 19 C 52 B 38 D 56

6. ABCD is a parallelogram with diagonals intersecting at E. If AE = 4x - 8 and EC = 36, find the value of x.

F 7 G 11 H 15.5 J 38

7. Find the values of the values of x and y so that the quadrilateral is a parallelogram.

A x = 27, y = 90 C x = 13, y = 90 B x = 27, y = 40 D x = 13, y = 40

8. Find the value of x so that the quadrilateral is a parallelogram.F 7 1 −

3 H 12

G 8 J 66

9. ABCD is a parallelogram with A(5, 4), B(-1, -2), and C(8, -2). Find the coordinates of D.

A D(-5, 4) B D(8, 2) C D(14, 4) D D(4, 1)

10. ABCD is a rectangle. If AB = 7x - 6 and CD = 5x + 30, find the value of x.

F 5 1 −

3 G 12 H 13 J 18

11. Which of the following is true for all rectangles? A The diagonals are perpendicular. B The consecutive angles are supplementary. C The opposite sides are supplementary. D The opposite angles are complementary.

6x - 6 3x + 30

(4x - 4)°

(3x + 23)°

(y - 60)°

1–3y °

38°

124°1

A D

B C

(2x - 10)°

(x + 30)°

(x + 70)° x°

(2x)°

(2x)°

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

Chapter 6 Test, Form 2B


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6

12. ABCD is a rectangle with B(-7, 3), C(5, 3), and D(5, -8). Find the coordinates of A.

F A(-8, -7) G A(-7, -8) H A(-5, -3) J A(-8, -5)

13. For rhombus GHJK, find m∠1. A 90 C 52 B 64 D 38

14. The diagonals of square ABCD intersect at E. If AE = 3x - 4 and BD = 10x - 48, find AC.

F 90 G 52 H 26 J 10

15. ABCD is an isosceles trapezoid with A(0, -1), B(-2, 3), and D(6, -1). Find the coordinates of C.

A C(6, 1) B C(9, 4) C C(2, 3) D C(8, 3)

16. For isosceles trapezoid MNOP, find m∠MNP. F 42 H 82 G 70 J 98

17. The length of one base of a trapezoid is 19 meters and the length of the median is 23 meters. Find the length of the other base.

A 15 m B 21 m C 27 m D 42 m

18 On a coordinate plane, the four corners of Ronald’s garden are located at (0, 2), (4, 6), (8, 2), and (4, -2). Which of the following most accurately describes the shape of Ronald’s garden?

F square H rhombus G rectangle J trapezoid

19. For kite WXYZ, find m∠W. A 106 C 212 B 148 D 360

20. ABCD is a parallelogram with coordinates A(4, 2), B(3, −1), C(−1, −1), andD(−1, 2). To prove that ABCD is a rhombus, you would plot the parallelogram on a coordinate plane and then find which of the following?

F measures of the angles H slopes of the diagonals G lengths of the diagonals J midpoints of the diagonals

Bonus The sum of the measures of the interior angles of a convex polygon is ten times the sum of the measures of its exterior angles. Find the number of sides of the polygon.

28°

42°

P

ON

M

128°

1

G K

JH

12.

13.

14.

15.

16.

17.

18.

19.

20.

B:

130° 18°

Chapter 6 Test, Form 2B (continued)


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6 SCORE

1. What is the sum of the interior angles of an octagonal box?

2. A convex pentagon has interior angles with measures (5x - 12)°, (2x + 100)°, (4x + 16)°, (6x + 15)°, and (3x + 41)°. Find the value of x.

3. If the measure of each interior angle of a regular polygon is 171, find the number of sides in the polygon.

4. In parallelogram ABCD, m∠1 = x + 12, and m∠2 = 6x - 18. Find m∠1.

5. Find the measure of each exterior angle of a regular 45-gon.

6. In parallelogram ABCD, m∠A = 58. Find m∠B.

7. Find the coordinates of the intersection of the diagonals of parallelogram XYZW with vertices X(2, 2), Y(3, 6), Z(10, 6), and W(9, 2).

8. Determine whether ABCD is a parallelogram. Justify your answer.

9. Determine whether the quadrilateral with vertices A(5, 7), B(1, -2), C(-6, -3), and D(2, 5) is a parallelogram. Use the slope formula.

10. For quadrilateral ABCD, the slope of −−

AB is 1 −

4 , the slope of

−−−

BC

is -

2 −

3 , and the slope of

−−−

CD is 1 −

4 . Find the slope of

−−−

DA so that

ABCD will be a parallelogram.

11. Given rectangle ABCD, find the value of x.

12. ABCD is a parallelogram and −−

AC � −−−

BD . Determine whether ABCD is a rectangle. Justify your answer.

13. ABCD is a rhombus with diagonals intersecting at E. If m∠ABC is three times m∠BAD, find m∠EBC.

A B

CD

(2x + 10)°

(3x - 30)°

A D

CB

12 C

D

A

B

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

Chapter 6 Test, Form 2C


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14. TUVW is a square with U(10, 2), V(8, 8), and W(2, 6). Find the coordinates of T.

15. For isosceles trapezoid MNOP, find m∠MNQ.

16. ABCD is a quadrilateral with vertices A(8, 3), B(6, 7), C(-1, 5), and D(-6, -1). Determine whether ABCD is a trapezoid. Justify your answer.

17. The length of the median of trapezoid EFGH is 13 feet. If the bases have lengths 2x + 4 and 10x - 50, find x.

18. ABCD is a kite, If RC = 10, and BD = 48, find CD.


19. A rectangle is always a parallelogram.

20. The diagonals of a rhombus are always perpendicular.

21. The diagonals of a square always bisect each other.

22. A trapezoid always has two congruent sides.

23. The median of a trapezoid is always parallel to the bases.

24. A kite has exactly two congruent angles.

25. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rectangle.

Bonus In parallelogram ABCD, AB = 2x - 7, BC = x + 3y, CD = x + y, and AD = 2x - y - 1. Find the values of x and y.

N

Q

O

PM 59°

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

Chapter 6 Test, Form 2C (continued)


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6 SCORE

1. Bruce is building a tabletop in the shape of an octagon. Find the sum of the external angles of the tabletop.

2. A convex octagon has interior angles with measures (x + 55)°, (3x + 20)°, 4x°, (4x - 10)°, (6x - 55)°, (3x + 52)°, 3x°, and (2x + 30)°. Find the value of x.

3. If the measure of each interior angle of a regular polygon is 176 find the number of sides in the polygon.

4. In parallelogram ABCD, m∠1 = x + 25, and m∠2 = 2x. Find m∠2.

5. Find the measure of each exterior angle of a regular 100-gon.

6. In parallelogram ABCD, m∠A = 63. Find m∠B.

7. Find the coordinates of the intersection of the diagonals of parallelogram XYZW with vertices X(3, 0), Y(3, 8), Z(-2, 6), and W(-2, -2).

8. Determine whether this quadrilateral is a parallelogram. Justify your answer.

9. Determine whether a quadrilateral with vertices A(5, 7), B(1, -1), C(-6, -3), and D(-2, 5) is a parallelogram. Use the slope formula.

10. If the slope of −−

AB is 1 −

2 , the slope of

−−−

BC is -4, and the slope of −−−

CD is 1 −

2 , find the slope of

−−−

DA so that ABCD is a parallelogram.

11. For rectangle ABCD, find the value of x.

12. ABCD is a parallelogram and m∠A = 90. Determine whether ABCD is a rectangle. Justify your answer.

B C

DA

(6x + 20)°

(3x - 2)°

1

2A D

CB

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

Chapter 6 Test, Form 2D


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13. ABCD is a rhombus with diagonals intersecting at E. If m∠ABC is four times m∠BAD, find m∠EBC.

14. PQRS is a square with Q(-2, 8), R(5, 7), and S(4, 0). Find the coordinates of P.

15. For isosceles trapezoid MNOP, find m∠MNQ.

16. ABCD is a quadrilateral with A(8, 21), B(10, 27), C(26, 26), and D(18, 2). Determine whether ABCD is a trapezoid. Justify your answer.

17. The length of the median of trapezoid EFGH is 17 centimeters. If the bases have lengths 2x + 4 and 8x - 50, find the value of x.

18. For kite ABCD, if RA = 15, and BD = 16, find AD.


19. A parallelogram always has four right angles.

20. The diagonals of a rhombus always bisect the angles.

21. A rhombus is always a square.

22. A rectangle is always a square.

23. The diagonals of an isosceles trapezoid are always congruent.

24. The median of a trapezoid always bisects the angles.

25. The diagonals of a kite are always perpendicular.

Bonus The measure of each interior angle of a regular polygon is 24 more than 38 times the measure of each exterior angle. Find the number of sides of the polygon.

M P

ON

Q74°

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

Chapter 6 Test, Form 2D (continued)


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6 SCORE

1. The sum of the interior angles of an animal pen is 900°. How many sides does the pen have?

2. A convex hexagon has interior angles with measures x°, (5x - 103)°, (2x + 60)°, (7x - 31)°, (6x - 6)°, and (9x - 100)°. Find the value of x and the measure of each angle.

3. Find the measure of each exterior angle of a regular 2x-gon.

4. For parallelogram ABCD, find m∠1.

5. ABCD is a parallelogram with diagonals that intersect each other at E. If AE = x2 and EC = 6x - 8, find all possible values of AC.

6. Determine whether the quadrilateral is a parallelogram. Justify your answer.

7. For quadrilateral ABCD, the slope of −−

AB is 2 −

3 and the slope

of −−−

BC is -2. Find the slopes of −−−

CD and −−−

DA so that ABCD will be a parallelogram.

8. In rectangle ABCD, find m∠1.

9. The diagonals of rhombus ABCD intersect at E. If m∠BAE = 2 −

3 (m∠ABE), find m∠BCD.

10. The diagonals of square ABCD intersect at E. If AE = 2, find the perimeter of ABCD.

11. For isosceles trapezoid ABCD, find AE.

12. Points G and H are midpoints of −−

AF and −−−

DE inregular hexagon ABCDEF. If AB = 6 find GH.

13. The vertices of trapezoid ABCD are A(10, -1), B(6, 6), C(-2, 6), and D(-8, -1). Find the length of the median.

B C

F

A DHG

E

A D

CB

E F60°

8

B C

DA(3x - 16)°

1(2x + 10)°

96° 31°1

D C

BA

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

Chapter 6 Test, Form 3


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6

14.

15.

16.

17.

18.

19.

20.

B:

14. Determine whether the quadrilateral ABCD with vertices A(0,−1), B(−4,−3), C(−5, 1), D(1, 7) is a kite. Justify your answer.

15. Determine whether the quadrilateral ABCD with vertices A(6, 2), B(2, 10), C(-6, 6), and D(-2, -2) is a rectangle. Justify your answer.

16. Determine whether quadrilateral ABCD with vertices A(1, 6), B(7, 6), C(2, -3), and D(-4, -3) is a parallelogram. Use the distance formula.

17. Find the value of x in kite EFGH.

For Questions 18 and 19, complete the two-column proof by supplying the missing information for each corresponding location.

Given: ABCD is a parallelogram.

−−−

BQ � −−−

DS , −−

PA � −−−

RC Prove: PQRS is a parallelogram.

Statements Reasons 1. ABCD is a �.

2. −−

AD � −−

CB

3. −−

PA � −−

RC

4. −−

PD � −−

RB

5. −−

AB � −−

CD

6. −−

BQ � −−

DS

7. −−

AQ � −−

CS

8. ∠B � ∠D, ∠A � ∠C

9. �QBR � �SDP, �PAQ � �RCS

10. −−

QP � −−

RS , −−

QR � −−

PS 11. PQRS is a parallelogram.

1. Given

2. (Question 18)

3. Given

4. Seg. Sub. Prop.

5. Opp. sides of a � are �.

6. Given

7. Seg. Sub. Prop.

8. Opp. � of a � are �.

9. SAS

10. CPCTC

11. (Question 19)

20. In isosceles trapezoid ABCD, AE = 2x + 5, EC = 3x - 12, and BD = 4x + 20. Find the value of x.

Bonus If three of the interior angles of a convex hexagon each measure 140, a fourth angle measures 84, and the measure of the fifth angle is 3 times the measure of the sixth angle, find the measure of the sixth angle.

(x + 5)°

x°(140 - x)°

B C

EDA

A P

R

D

C

SQ

B

Chapter 6 Test, Form 3 (continued)


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6 SCORE Chapter 6 Extended-Response Test

Demonstrate your knowledge by giving a clear, concise solution to each problem. Be sure to include all relevant drawings and justify your answers. You may show your solution in more than one way or investigate beyond the requirements of the problem.

1. a. Draw a regular convex polygon and a convex polygon that is not regular, each with the same number of sides.

b. Label the measures of each exterior angle on your figures.

c. Find the sum of the exterior angles for each figure. What conjecture can be made?

2. Draw a rectangle. Connect the midpoints of the consecutive sides. What type of quadrilateral is formed? How do you know?

3. Draw an example to show why one pair of opposite sides congruent and the other pair of opposite sides parallel is not sufficient to form a parallelogram.

4. a. Name a property that is true for a square and not always true for a rectangle.

b. Name a property that is true for a square and not always true for a rhombus.

c. Name a property that is true for a rectangle and not always true for a parallelogram.


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6 SCORE

1. Find the coordinates of X if V(0.5, 5) is the midpoint of −−−

UX with U(15, 21). (Lesson 1-3)

A (-14, -11) C (0, 0) B (7.75, 22.5) D (15.5, -5)

2. Which of the following are possible measures for vertical angles G and H? (Lesson 2-8)

F m∠G = 125 and m∠H = 55 G m∠G = 125 and m∠H = 125 H m∠G = 55 and m∠H = 45 J m∠G = 55 and m∠H = 152.5

3. Determine which lines are parallel. (Lesson 3-5)

A � �� NS ‖ � �� PT C � �� QR ‖ � �� ST

B � �� NP ‖ � �� ST D � �� NP ‖ � �� QR

4. Find the coordinates of B, the midpoint of −−

AC , if A(2a, b) and C(0, 2b). (Lesson 4-8)

F (2a, 2b) G (a, b) H (a, 3 −

2 b) J ( 3 −

2 a, b)

5. If −−

RV is an angle bisector, find m∠UVT. (Lesson 5-1)

A 10 C 68 B 34 D 136

6. Find the slope of the line that passes through points A(-7, 14) and B(5, -2). (Lesson 3-3)

F - 4 −

3 G -

3 −

4 H 3 −

4 J 4 −

3

7. Which statement ensures that quadrilateral QRST is a parallelogram? (Lesson 6-3)

A ∠Q � ∠S C −−−

QT ‖ −−

RS B

−−−

QR � −−

TS and −−−

QR ‖ −−

TS D m∠Q + m∠S = 180

Q R

ST

(2y + 14)°

(4y - 6)°

T VS

R

U

67°

113°

65°

N P

ST

RQ

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

Standardized Test Practice(Chapters 1–6)

Part 1: Multiple Choice

Instructions: Fill in the appropriate circle for the best answer.


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8. What is the equation of the line that contains (-12, 9) and is perpendicular to the line y =

2 −

3 x + 5? (Lesson 3-4)

F y = - 3 −

2 x - 9 H y = - 2 −

3 x - 1

G y = 3 −

2 x - 1 J y = 2 −

3 x + 17

9. Which of the following theorems can be used to prove �ABC � �DEC? (Lesson 4-5)

A SSS C SAS B AAS D ASA

10. What is the value of x? (Lesson 6-6)

F 2 H 5.5 G 4 J 7

11. For �ABC, AB = 6 and BC = 17. Which of the following is a possible length for

−−−

AC ? (Lesson 5-3)

A 5 B 9 C 13 D 24

12. What is m∠T in kite STVW? F 100 H 95 G 130 J 260

A

D

B C

E

8. F G H J

9. A B C D

10. F G H J

11. A B C D

12. F G H J

13. If �UVW is an isosceles triangle, −−−

UV � −−−

WU , UV = 16b - 40, VW = 6b, and WU = 10b + 2, find the value of b. (Lesson 4-1)

14. Find the sum of the measures of the interior angles for a convex heptagon. (Lesson 6-1)

13.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

14.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

Standardized Test Practice (continued)

4x - 4

16

20

Part 2: Gridded Response

Instructions: Enter your answer by writing each digit of the answer

in a column box and then shading in the appropriate circle that

corresponds to that entry.

85° 15°


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6

15. A polygon has six congruent sides. Lines containing two of its sides contain points in its interior. Name the polygon by its number of sides, and then classify it as convex or concave and regular or irregular. (Lesson 1-6)

16. If −−

RT � −−−

QM and RT = 88.9 centimeters, find QM. (Lesson 2-7)

17. Which segment is the shortest segment from D to

�

�� JM ? (Lesson 5-2)

18. If �ABC � �WXY, AB = 72, BC = 65, CA = 13, XY = 7x - 12, and WX = 19y + 34, find the values of x and y. (Lesson 4-3)

19. Freda bought two bells for just over $90 before tax. State the assumption you would make to write an indirect proof to show that at least one of the bells costs more than $45. (Lesson 5-4)

20. The area of the base of a cylinder is 5 cm2 and the height of the cylinder is 8 cm. Find the volume of the cylinder. (Lesson 1-7)

21. JKLM is a kite. Complete each statement. (Lesson 6-6)

a. −−−

MJ �

b. −−−

MK ⊥

c. m∠L = m∠

D

J K L H M

15.

16.

17.

18.

19.

20.

21. a.

b.

c.

Part 3: Short Response

Instructions: Place your answers in the space provided.

6

6

Standardized Test Practice (continued)


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6 SCORE

1. Use a protractor to classify �UVW, �UWX, and �XWY as acute, equiangular, obtuse, or right.

2. In the figure, ∠1 � ∠2. Find the measures of the numbered angles.

3. Name the corresponding congruent sides for �AFP � �STX.

4. Determine whether �ABC � �PQR given A(2, -7), B(5, 3), C(-4, 6), P(8, -1), Q(11, 9), and R(2, 12).

5. In the figure, −−

LK bisects ∠JKM and ∠KLJ � ∠KLM. Determine which theorem or postulate can be used to prove that �JKL � �MKL.

6. Triangle ABC is isosceles with AB = BC. Name a pair of congruent angles in this triangle.

7. For kite WXYZ, find m∠Z.

For Questions 8 and 9, refer to the figure.

8. Find the value of a and m∠ZWT if −−−

ZW is an altitude of �XYZ, m∠ZWT = 3a + 5, and m∠TWY = 5a + 13.

9. Determine which angle has the greatest measure: ∠YWZ, ∠WZY, or ∠ZYW.

10. Mr. Ramirez bought a stove and a dishwasher for just over $1206. State the assumption you would make to start an indirect proof to show that at least one of the appliances cost more than $603.

X W Y

T

Z

K L

J

M

65°

70°

110°13

2

D

E F H

G

V

U

W

X

Y

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

Unit 2 Test(Chapters 4 – 6)

43° 95°


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11. Determine whether 128 feet, 136 feet, and 245 feet can be the lengths of the sides of a triangle.

12. Write an inequality to describe the possible values of x.

13. The measure of an interior angle of a regular polygon is 140. Find the number of sides in the polygon.

14. For parallelogram JKMH, find m∠JHK, m∠HMK, and the value of x.

15. Determine whether the vertices of quadrilateral DEFG form a parallelogram given D(-3, 5), E(3, 6), F(-1, 0), and G(6, 1).

16. For rectangle WXYZ with diagonals −−−

WY and −−

XZ ,WY = 3d + 4 and XZ = 4d - 1, find the value of d.

17. If m∠BEC = 9z + 45 in rhombus ABCD, find the value of z.

18. In trapezoid HJLK, M and N are midpoints of the legs. Find KL.

19. Prove that quadrilateral PQRS isNOT a parallelogram.

JH

K L

NM

45

28

A C

B

D

E

J

K

M

H20°

3x + 8

7x - 2452°

2x - 512 12

14

11

14

41˚38˚

11.

12.

13.

14.

15.

16.

17.

18.

19.

x

y

-1

12345

1 2 3 4-2-3-4

Unit 2 Test (continued)


An

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Chapter 6 A1 Glencoe Geometry

Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.

Chapter Resources

NA

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Ch

ap

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and

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7.

If a

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10.

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has

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11.

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OC

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Lesson 6-1


NA

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D

Ch

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6

5

Gle

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Geo

met

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Stud

y G

uide

and

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An

gle

s o

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oly

go

ns

Poly

go

n In

teri

or

An

gle

s Su

m T

he

segm

ents

th

at c

onn

ect

the

non

con

secu

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f a

poly

gon

are

cal

led

dia

gon

als.

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win

g al

l of

th

e di

agon

als

from

on

e ve

rtex

of

an n

-gon

sep

arat

es t

he

poly

gon

in

to n

- 2

tri

angl

es. T

he

sum

of

the

mea

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s of

th

e in

teri

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ngl

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f th

e po

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n c

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x p

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on h

as

13 s

ides

. Fin

d t

he

sum

of

the

mea

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s of

th

e in

teri

or a

ngl

es.

(n -

2)

� 1

80 =

(13

- 2

) � 1

80=

11

�

180

= 1

980

T

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mea

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of

an

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0. F

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Exer

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ind

th

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m o

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th

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pol

ygon

.

7. 1

50

8. 1

60

9. 1

75

10. 1

65

11. 1

44

12. 1

35

13. F

ind

the

valu

e of

x.

6 -1

7x°

( 4x

+ 1

0)°

( 4x

+ 5

) ° ( 6x

+ 1

0)°

( 5x

- 5

) °

AB

C

D

E

Po

lyg

on

In

teri

or

An

gle

Su

m T

he

ore

mT

he

su

m o

f th

e in

terio

r a

ng

le m

ea

su

res o

f a

n n

-sid

ed

co

nve

x p

oly

go

n is (

n -

2

) · 1

80

.

Exam

ple

1Ex

amp

le 2

1

440

2520

5040

1

080

1800

5940

1

2

18

72

2

4

10

8

2

0

001_

042_

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OC

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M

Answers (Anticipation Guide and Lesson 6-1)

A01_A21_GEOCRMC06_890515.indd A1A01_A21_GEOCRMC06_890515.indd A1 5/20/08 6:04:31 PM5/20/08 6:04:31 PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

6

Gle

ncoe

Geo

met

ry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

An

gle

s o

f P

oly

go

ns

Poly

go

n E

xter

ior

An

gle

s Su

m T

her

e is

a s

impl

e re

lati

onsh

ip a

mon

g th

e ex

teri

or a

ngl

es o

f a

con

vex

poly

gon

.

F

ind

th

e su

m o

f th

e m

easu

res

of t

he

exte

rior

an

gles

, on

e at

ea

ch v

erte

x, o

f a

con

vex

27-g

on.

For

an

y co

nve

x po

lygo

n, t

he

sum

of

the

mea

sure

s of

its

ext

erio

r an

gles

, on

e at

eac

h

vert

ex, i

s 36

0.

F

ind

th

e m

easu

re o

f ea

ch e

xter

ior

angl

e of

regu

lar

hex

agon

AB

CD

EF

.

Th

e su

m o

f th

e m

easu

res

of t

he

exte

rior

an

gles

is

360

and

a h

exag

on

has

6 a

ngl

es. I

f n

is

the

mea

sure

of

each

ext

erio

r an

gle,

th

en 6n

= 3

60 n

= 6

0T

he

mea

sure

of

each

ext

erio

r an

gle

of a

reg

ula

r h

exag

on i

s 60

.

Exer

cise

sF

ind

th

e su

m o

f th

e m

easu

res

of t

he

exte

rior

an

gles

of

each

con

vex

pol

ygon

.

1. d

ecag

on

2. 1

6-go

n

3. 3

6-go

n

Fin

d t

he

mea

sure

of

each

ext

erio

r an

gle

for

each

reg

ula

r p

olyg

on.

4. 1

2-go

n

5. h

exag

on

6. 2

0-go

n

7. 4

0-go

n

8. h

epta

gon

9.

12-

gon

10. 2

4-go

n

11. d

odec

agon

12

. oct

agon

AB

C

DE

F

6 -1

Po

lyg

on

Ex

teri

or

An

gle

Su

m T

he

ore

m

Th

e s

um

of

the

exte

rio

r a

ng

le m

ea

su

res o

f a

co

nve

x p

oly

go

n,

on

e a

ng

le

at

ea

ch

ve

rte

x,

is 3

60

.

Exam

ple

1

Exam

ple

2

3

60

360

360

3

0

60

18

9

5

1.4

3

0

1

5

30

45

001_

042_

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OC

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515.

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:38:

28P

M


NA

ME

DA

TE

PE

RIO

D

Lesson 6-1

Ch

ap

ter

6

7

Gle

ncoe

Geo

met

ry

Skill

s Pr

acti

ceA

ng

les o

f P

oly

go

ns

Fin

d t

he

sum

of

the

mea

sure

s of

th

e in

teri

or a

ngl

es o

f ea

ch c

onve

x p

olyg

on.

1. n

onag

on

2. h

epta

gon

3.

dec

agon

Th

e m

easu

re o

f an

in

teri

or a

ngl

e of

a r

egu

lar

pol

ygon

is

give

n. F

ind

th

e n

um

ber

of

sid

es i

n t

he

pol

ygon

.

4. 1

08

5. 1

20

6. 1

50

Fin

d t

he

mea

sure

of

each

in

teri

or a

ngl

e.

7.

8.

9.

10.

Fin

d t

he

mea

sure

s of

eac

h i

nte

rior

an

gle

of e

ach

reg

ula

r p

olyg

on.

11. q

uad

rila

tera

l 12

. pen

tago

n

13. d

odec

agon

Fin

d t

he

mea

sure

s of

eac

h e

xter

ior

angl

e of

eac

h r

egu

lar

pol

ygon

.

14. o

ctag

on

15. n

onag

on

16. 1

2-go

n

6-1

x°

x°

( 2x

- 1

5)°

( 2x

- 1

5)°

AB

CD

( 2x)

°

( 2x

+ 2

0)°

( 3x

- 1

0)°

( 2x

- 1

0)°

LM

NP

(2x

+16

)°

(x+

14)°

(x+

14)°

(2x

+16

)°(7

x)°

(7x)

°

(4x)

°(4

x)°

(7x)

°(7

x)°

1

260

900

1440

5

6

1

2

m

∠A

= 1

15,

m∠

B =

65,

m

∠L =

100,

m∠

M =

110,

m∠

C =

115,

m∠

D =

65

m

∠N

= 7

0,

m∠

P =

80

m

∠S

= 1

16,

m∠

T =

116,

m∠

D =

140,

m∠

E =

140,

m

∠W

= 6

4,

m∠

U =

64

m∠

I =

80,

m∠

F =

80,

m∠

H =

140,

m∠

G =

140

9

0,

90

108,

72

150,

30

4

5

40

30

001_

042_

GE

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C06

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515.

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17/0

86:

05:5

4P

M

Answers (Lesson 6-1)


An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

8

Gle

ncoe

Geo

met

ry

Fin

d t

he

sum

of

the

mea

sure

s of

th

e in

teri

or a

ngl

es o

f ea

ch c

onve

x p

olyg

on.

1. 1

1-go

n

2. 1

4-go

n

3. 1

7-go

n

Th

e m

easu

re o

f an

in

teri

or a

ngl

e of

a r

egu

lar

pol

ygon

is

give

n. F

ind

th

e n

um

ber

of

sid

es i

n t

he

pol

ygon

.

4. 1

44

5. 1

56

6. 1

60

Fin

d t

he

mea

sure

of

each

in

teri

or a

ngl

e.

7.

8.

Fin

d t

he

mea

sure

s of

an

ext

erio

r an

gle

and

an

in

teri

or a

ngl

e gi

ven

th

e n

um

ber

of

sid

es o

f ea

ch r

egu

lar

pol

ygon

. Rou

nd

to

the

nea

rest

ten

th, i

f n

eces

sary

.

9. 1

6 10

. 24

11. 3

0

12. 1

4 13

. 22

14. 4

0

15. C

RY

STA

LLO

GR

APH

Y C

ryst

als

are

clas

sifi

ed a

ccor

din

g to

sev

en c

ryst

al s

yste

ms.

Th

e ba

sis

of t

he

clas

sifi

cati

on i

s th

e sh

apes

of

the

face

s of

th

e cr

ysta

l. T

urq

uoi

se b

elon

gs t

o th

e tr

icli

nic

sys

tem

. Eac

h o

f th

e si

x fa

ces

of t

urq

uoi

se i

s in

th

e sh

ape

of a

par

alle

logr

am.

Fin

d th

e su

m o

f th

e m

easu

res

of t

he

inte

rior

an

gles

of

one

such

fac

e.

Prac

tice

An

gle

s o

f P

oly

go

ns

6 -1

x°

( 2x

+ 1

5)°

( 3x

- 2

0)°

( x+

15)

°

JK

MN

(6x

-4)

°

(2x

+8)

°(6

x-

4)°

(2x

+8)

°

m

∠R

= 1

28,

m∠

S =

52

m∠

T =

128,

m∠

U =

52

1

620

2160

2700

1

0

15

18

m

∠J =

115,

m∠

K =

130,

m∠

M =

50,

m∠

N =

65

1

57.5

, 22.5

1

65,

15

168,

12

1

54.3

, 25.7

1

63.6

, 16.4

1

71,

9

3

60

001_

042_

GE

OC

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C06

_890

515.

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811

:38:

39P

M

Lesson 6-1


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

9

Gle

ncoe

Geo

met

ry

Wor

d Pr

oble

m P

ract

ice

An

gle

s o

f P

oly

go

ns

La T

ribun

a

1. A

RC

HIT

ECTU

RE

In t

he

Uff

izi

gall

ery

in F

lore

nce

, Ita

ly, t

her

e is

a r

oom

bu

ilt

by B

uon

tale

nti

cal

led

the

Tri

bun

e (L

a T

ribu

na

in I

tali

an).

Th

is r

oom

is

shap

ed

like

a r

egu

lar

octa

gon

.

Wh

at a

ngl

e do

con

secu

tive

wal

ls o

f th

e T

ribu

ne

mak

e w

ith

eac

h o

ther

?

2. B

OX

ES J

asm

ine

is d

esig

nin

g bo

xes

she

wil

l u

se t

o sh

ip h

er je

wel

ry. S

he

wan

ts

to s

hap

e th

e bo

x li

ke a

reg

ula

r po

lygo

n.

In o

rder

for

th

e bo

xes

to p

ack

tigh

tly,

sh

e de

cide

s to

use

a r

egu

lar

poly

gon

th

at

has

th

e pr

oper

ty t

hat

th

e m

easu

re o

f it

s in

teri

or a

ngl

es i

s h

alf

the

mea

sure

of

its

exte

rior

an

gles

. Wh

at r

egu

lar

poly

gon

sh

ould

sh

e u

se?

3. T

HEA

TER

A t

hea

ter

floo

r pl

an i

s sh

own

in

th

e fi

gure

. Th

e u

pper

fiv

e si

des

are

part

of

a re

gula

r do

deca

gon

.

Fin

d m

∠1.

4. A

RC

HEO

LOG

Y A

rch

eolo

gist

s u

nea

rth

ed

part

s of

tw

o ad

jace

nt

wal

ls o

f an

an

cien

t ca

stle

.

Bef

ore

it w

as u

nea

rth

ed, t

hey

kn

ew f

rom

an

cien

t te

xts

that

th

e ca

stle

was

sh

aped

li

ke a

reg

ula

r po

lygo

n, b

ut

nob

ody

knew

h

ow m

any

side

s it

had

. Som

e sa

id 6

, ot

her

s 8,

an

d so

me

even

sai

d 10

0. F

rom

th

e in

form

atio

n i

n t

he

figu

re, h

ow m

any

side

s di

d th

e ca

stle

rea

lly

hav

e?

5. P

OLY

GO

N P

ATH

In M

s. R

icke

ts’ m

ath

cl

ass,

stu

den

ts m

ade

a “p

olyg

on p

ath

” th

at c

onsi

sts

of r

egu

lar

poly

gon

s of

3, 4

, 5,

an

d 6

side

s jo

ined

tog

eth

er a

s sh

own

.

a. F

ind

m∠

2 an

d m

∠5.

b.

Fin

d m

∠3

and

m∠

4.

c. W

hat

is

m∠

1?

6 -1

21

34

5

stag

e1

24˚

1

35°

a

n e

qu

ilate

ral

tria

ng

le

1

20

1

5

90 a

nd

60

162 a

nd

132

96

001_

042_

GE

OC

RM

C06

_890

515.

indd

94/

11/0

811

:38:

45P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

10

G

lenc

oe G

eom

etry

Enri

chm

ent

Cen

tral

An

gle

s o

f R

eg

ula

r P

oly

go

ns

You

hav

e le

arn

ed a

bou

t th

e in

teri

or a

nd

exte

rior

an

gles

of

a po

lygo

n. R

egu

lar

poly

gon

s al

so h

ave

cen

tral

an

gles

. A c

entr

al a

ngl

e is

mea

sure

d fr

om t

he

cen

ter

of

the

poly

gon

.

Th

e ce

nte

r of

a p

olyg

on i

s th

e po

int

equ

idis

tan

t fr

om a

ll o

f

the

vert

ices

of

the

poly

gon

, ju

st a

s th

e ce

nte

r of

a c

ircl

e is

th

e po

int

equ

idis

tan

t fr

om a

ll o

f th

e po

ints

on

th

e ci

rcle

. Th

e ce

ntr

al a

ngl

e is

th

e an

gle

draw

n w

ith

th

e ve

rtex

at

the

cen

ter

of t

he

circ

le a

nd

the

side

s of

an

gle

draw

n t

hro

ugh

co

nse

cuti

ve v

erti

ces

of t

he

poly

gon

. On

e of

th

e ce

ntr

al

angl

es t

hat

can

be

draw

n i

n t

his

reg

ula

r h

exag

on i

s ∠

AP

B.

You

may

rem

embe

r fr

om m

akin

g ci

rcle

gra

phs

that

th

ere

are

360°

aro

un

d th

e ce

nte

r of

a c

ircl

e.

1. B

y u

sin

g lo

gic

or b

y dr

awin

g sk

etch

es, f

ind

the

mea

sure

of

the

cen

tral

an

gle

of e

ach

re

gula

r po

lygo

n.

2. M

ake

a co

nje

ctu

re a

bou

t h

ow t

he

mea

sure

of

a ce

ntr

al a

ngl

e of

a r

egu

lar

poly

gon

rel

ates

to

th

e m

easu

res

of t

he

inte

rior

an

gles

an

d ex

teri

or a

ngl

es o

f a

regu

lar

poly

gon

.

3. C

HA

LLEN

GE

In o

btu

se �

AB

C,

−−

−

B

C i

s th

e lo

nge

st s

ide.

−−

A

C i

s al

so a

sid

e of

a

21-s

ided

reg

ula

r po

lygo

n.

−−

A

B i

s al

so a

sid

e of

a 2

8-si

ded

regu

lar

poly

gon

. Th

e 21

-sid

ed r

egu

lar

poly

gon

an

d th

e 28

-sid

ed r

egu

lar

poly

gon

hav

e th

e sa

me

cen

ter

poin

t P

. Fin

d n

if

−−

−

B

C i

s a

side

of

a n

-sid

ed r

egu

lar

poly

gon

th

at h

as

cen

ter

poin

t P

.

(H

int:

Ske

tch

a c

ircl

e w

ith

cen

ter

P a

nd

plac

e po

ints

A, B

, an

d C

on

th

e ci

rcle

.)

6 -1

A

B

P

r

1

20

9

0

72

6

0

ab

ou

t 51.4

3

4

5

T

he m

easu

re o

f th

e e

xte

rio

r an

gle

eq

uals

th

e m

easu

re o

f th

e c

en

tral

an

gle

. T

he c

en

tral

an

gle

is s

up

ple

men

tary

to

in

teri

or

an

gle

.

n

= 1

2

001_

042_

GE

OC

RM

C06

_890

515.

indd

104/

11/0

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:38:

50P

M

Lesson 6-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

11

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

Para

llelo

gra

ms

Sid

es a

nd

An

gle

s o

f Pa

ralle

log

ram

s A

qu

adri

late

ral

wit

h

both

pai

rs o

f op

posi

te s

ides

par

alle

l is

a p

aral

lelo

gram

. Her

e ar

e fo

ur

impo

rtan

t pr

oper

ties

of

para

llel

ogra

ms.

If

AB

CD

is

a p

aral

lelo

gram

, fin

d t

he

valu

e of

eac

h v

aria

ble

.−

−

A

B a

nd

−−

−

C

D a

re o

ppos

ite

side

s, s

o −

−

A

B �

−−

−

C

D .

2a =

34

a =

17

∠A

an

d ∠

C a

re o

ppos

ite

angl

es, s

o ∠

A �

∠C

. 8b

= 1

12 b

= 1

4

Exer

cise

sF

ind

th

e va

lue

of e

ach

var

iab

le.

1.

2.

3.

4.

5.

6.

PQ

RS

BA

DC

34

2a8b

°

112°

4y°

3x°

8y 88

6x°

72x

30x

150

2y

6 -2

If P

QR

S i

s a

pa

rall

elo

gra

m,

the

n

If a

qu

ad

rila

tera

l is

a p

ara

llelo

gra

m,

the

n its

op

po

site

sid

es a

re c

on

gru

en

t.

−−

−

P

Q �

−−

S

R a

nd

−−

P

S �

−

−−

Q

R

If a

qu

ad

rila

tera

l is

a p

ara

llelo

gra

m,

the

n its

op

po

site

an

gle

s a

re c

on

gru

en

t.

∠P

� ∠

R a

nd

∠S

� ∠

Q

If a

qu

ad

rila

tera

l is

a p

ara

llelo

gra

m,

the

n its

co

nse

cu

tive

an

gle

s a

re

su

pp

lem

en

tary

.

∠P

an

d ∠

S a

re s

up

ple

me

nta

ry;

∠S

an

d ∠

R a

re s

up

ple

me

nta

ry;

∠R

an

d ∠

Q a

re s

up

ple

me

nta

ry;

∠Q

an

d ∠

P a

re s

up

ple

me

nta

ry.

If a

pa

ralle

log

ram

ha

s o

ne

rig

ht

an

gle

,

the

n it

ha

s f

ou

r rig

ht

an

gle

s.

If m

∠P

= 9

0,

the

n m

∠Q

= 9

0,

m∠

R =

90

, a

nd

m∠

S =

90

.

Exam

ple

3y 12

6x 60°55

°5x

°

2y°

6x°

12x°

3y°

x

= 3

0;

y =

22.5

x =

15;

y =

11

x

= 2

; y =

4

x =

10;

y =

40

x

= 1

3;

y =

32.5

x =

5;

y =

180

001_

042_

GE

OC

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C06

_890

515.

indd

114/

11/0

811

:38:

54P

M

Answers (Lesson 6-1 and Lesson 6-2)


An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

12

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Para

llelo

gra

ms

Dia

go

nal

s o

f Pa

ralle

log

ram

s T

wo

impo

rtan

t pr

oper

ties

of

pa

rall

elog

ram

s de

al w

ith

th

eir

diag

onal

s.

F

ind

th

e va

lue

of x

an

d y

in

par

alle

logr

am A

BC

D.

Th

e di

agon

als

bise

ct e

ach

oth

er, s

o A

E =

CE

an

d D

E =

BE

. 6x

= 2

4 4y

= 1

8 x

= 4

y

= 4

.5

Exer

cise

sF

ind

th

e va

lue

of e

ach

var

iab

le.

1.

2.

3.

4.

5.

6.

CO

OR

DIN

ATE

GEO

MET

RY

Fin

d t

he

coor

din

ates

of

the

inte

rsec

tion

of

the

dia

gon

als

of �

AB

CD

wit

h t

he

give

n v

erti

ces.

7. A

(3, 6

), B

(5, 8

), C

(3, −

2), a

nd

D(1

, −4)

8.

A(−

4, 3

), B

(2, 3

), C

(−1,

−2)

, an

d D

(−7,

−2)

9. P

RO

OF

Wri

te a

par

agra

ph p

roof

of

the

foll

owin

g.

G

iven

: �A

BC

D

P

rove

: �A

ED

� �

BE

C

AB

CD

P

AB

E

CD

6x

1824

4y

3x4y

812

4x

2y28

3x

30°

10

y

AB

CD

E

2x°60

°4y

°

3x°

2y12

6 -2

If A

BC

D i

s a

pa

rall

elo

gra

m,

the

n

If a

qu

ad

rila

tera

l is

a p

ara

llelo

gra

m,

the

n

its d

iag

on

als

bis

ect

ea

ch

oth

er.

AP

= P

C a

nd D

P =

PB

If a

qu

ad

rila

tera

l is

a p

ara

llelo

gra

m,

the

n e

ach

dia

go

na

l se

pa

rate

s t

he

pa

ralle

log

ram

into

tw

o c

on

gru

en

t tr

ian

gle

s.

�A

CD

�

�

CA

B a

nd

�

AD

B �

�

CB

D

Exam

ple

x

17

4y

x

= 4

; y =

2

x =

7;

y =

14

x =

15;

y =

7.5

x

= 3

1

−

3 ;

y =

10 √

�

3

x =

15;

y =

6 √

�

2

x =

15;

y =

√ �

�

241

D

iag

on

als

of

a p

ara

lle

log

ram

bis

ec

t e

ac

h o

the

r, s

o −

−

AE

� −

−

CE

an

d −

−

BE

� −

−

DE

. O

pp

os

ite

sid

es

of

a p

ara

lle

log

ram

are

co

ng

rue

nt,

th

ere

fore

−−

AD

� −

−

BC

. B

ec

au

se

co

rre

sp

on

din

g p

art

s o

f th

e t

wo

tri

an

gle

s a

re c

on

gru

en

t, t

he

tr

ian

gle

s a

re c

on

gru

en

t b

y S

SS

.

(

3,

2)

(-

2.5

, 0.5

)

001_

042_

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C06

_890

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12/0

85:

38:2

8A

M

Lesson 6-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

13

G

lenc

oe G

eom

etry

Skill

s Pr

acti

ceP

ara

llelo

gra

ms

ALG

EBR

AF

ind

th

e va

lue

of e

ach

var

iab

le.

1.

2.

3.

4.

5.

6.

CO

OR

DIN

ATE

GEO

MET

RY

Fin

d t

he

coor

din

ates

of

the

inte

rsec

tion

of

the

dia

gon

als

of �

HJ

KL

wit

h t

he

give

n v

erti

ces.

7. H

(1, 1

), J

(2, 3

), K

(6, 3

), L

(5, 1

) 8.

H(-

1, 4

), J

(3, 3

), K

(3, -

2), L

(-1,

-1)

9. P

RO

OF

Wri

te a

par

agra

ph p

roof

of

the

theo

rem

Con

secu

tive

an

gles

in

a p

aral

lelo

gram

ar

e su

pple

men

tary

.

DC

BA

6-2

x-

2y

+1

1926

x°44

°

y°

4x -2

3y-

1

y+5

x +10

2b-

1b

+3

3a-

5

2a

a+

146a

+4

10b

+19b

+8

(x+

8)°

(3x)

°

(y+

9)°

a

= 5

, b

= 4

x =

136,

y =

44

(

3.5

, 2)

(1

, 1)

G

iven

: �

AB

CD

P

rove:

∠A

an

d ∠

B a

re s

up

ple

men

tary

.∠

B a

nd

∠C

are

su

pp

lem

en

tary

.∠

C a

nd

∠D

are

su

pp

lem

en

tary

.∠

D a

nd

∠A

are

su

pp

lem

en

tary

.

P

roo

f: W

e a

re g

ive

n �

AB

CD

, s

o w

e k

no

w t

ha

t −

−

AB

||

−−

CD

an

d −

−

BC

||

−−

DA

by

the

de

fin

itio

n o

f a

pa

rall

elo

gra

m.

We

als

o k

no

w t

ha

t if

tw

o p

ara

lle

l li

ne

s a

re

cu

t b

y a

tra

ns

ve

rsa

l, t

he

n c

on

se

cu

tiv

e i

nte

rio

r a

ng

les

are

su

pp

lem

en

tary

.

So

, ∠

A a

nd

∠B

, ∠

B a

nd

∠C

, ∠

C a

nd

∠D

, a

nd

∠D

an

d ∠

A a

re p

air

s o

f

su

pp

lem

en

tary

an

gle

s.

x

= 4

3,

y =

12

0

x =

4,

y =

3

x

= 2

1,

y =

25

a =

2,

b =

7

001_

042_

GE

OC

RM

C06

_890

515.

indd

135/

17/0

86:

06:3

9P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

14

G

lenc

oe G

eom

etry

Prac

tice

Para

llelo

gra

ms

ALG

EBR

AFi

nd

th

e va

lue

of

each

var

iab

le.

1.

2.

3.

4.

ALG

EBR

A U

se �

RS

TU

to

fin

d e

ach

mea

sure

or

valu

e.

5. m

∠R

ST

=

6.

m∠

ST

U=

7. m

∠T

UR

=

8.

b=

CO

OR

DIN

ATE

GEO

MET

RY

Fin

d t

he

coor

din

ates

of

the

inte

rsec

tion

of

the

dia

gon

als

of �

PR

YZ

wit

h t

he

give

n v

erti

ces.

9. P

(2, 5

), R

(3, 3

), Y

(-2,

-3)

, Z(-

3, -

1)

10. P

(2, 3

), R

(1, -

2), Y

(-5,

-7)

, Z(-

4, -

2)

11. P

RO

OF

Wri

te a

par

agra

ph p

roof

of

the

foll

owin

g.

Giv

en: �

PR

ST

an

d �

PQ

VU

Pro

ve: ∠

V �

∠S

12. C

ON

STR

UC

TIO

N M

r. R

odri

quez

use

d th

e pa

rall

elog

ram

at

the

righ

t to

de

sign

a h

erri

ngb

one

patt

ern

for

a p

avin

g st

one.

He

wil

l u

se t

he

pavi

ng

ston

e fo

r a

side

wal

k. I

f m

∠1

is 1

30, f

ind

m∠

2, m

∠3,

an

d m

∠4.

TU

SR

B30

°23

4b-

1

25°

PR

S

UVQ

T

12

34

6 -2

(4x)

°

(y+

10)°

(2y-

40)°

4x

5y-

83y

+1

x+6

y +3

15

x-3

12

b+1

2b

a+2

3a-

4

125

(

0,

1)

(-

1.5

, -

2)

P

roo

f: W

e a

re g

iven

�P

RS

T a

nd

�P

QV

U.

Sin

ce o

pp

osit

e a

ng

les o

f a

para

llelo

gra

m a

re c

on

gru

en

t, ∠

P �

∠V

an

d ∠

P �

∠S

. S

ince c

on

gru

en

ce

of

an

gle

s i

s t

ran

sit

ive,

∠V

� ∠

S b

y t

he T

ran

sit

ive P

rop

ert

y o

f co

ng

ruen

ce.

5

0,

130,

50

a

= 3

, b

= 1

x =

30

˚,

y =

50

˚

x

= 1

8,

y =

9

x =

2,

y =

4.5

125

6

55

001_

042_

GE

OC

RM

C06

_890

515.

indd

144/

11/0

811

:39:

16P

M

Lesson 6-2


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

15

G

lenc

oe G

eom

etry

1. W

ALK

WA

Y A

wal

kway

is

mad

e by

ad

join

ing

fou

r pa

rall

elog

ram

s as

sh

own

.

Are

th

e en

d se

gmen

ts a

an

d e

para

llel

to

each

oth

er?

Exp

lain

.

2. D

ISTA

NC

E F

our

frie

nds

liv

e at

th

e fo

ur

corn

ers

of a

blo

ck s

hap

ed l

ike

a pa

rall

elog

ram

. Gra

cie

live

s 3

mil

es a

way

fr

om K

enn

y. H

ow f

ar a

part

do

Ter

esa

and

Tra

vis

live

fro

m e

ach

oth

er?

3. S

OC

CER

Fou

r so

ccer

pla

yers

are

loc

ated

at

th

e co

rner

s of

a p

aral

lelo

gram

. T

wo

of t

he

play

ers

in o

ppos

ite

corn

ers

are

the

goal

ies.

In

ord

er f

or g

oali

e A

to

be

able

to

see

the

thre

e ot

her

s, s

he

mu

st

be a

ble

to s

ee a

cer

tain

an

gle

x in

her

fi

eld

of v

isio

n.

Wh

at a

ngl

e do

es t

he

oth

er g

oali

e h

ave

to

be a

ble

to s

ee i

n o

rder

to

keep

an

eye

on

th

e ot

her

th

ree

play

ers?

4. V

ENN

DIA

GR

AM

S M

ake

a V

enn

di

agra

m s

how

ing

the

rela

tion

ship

be

twee

n s

quar

es, r

ecta

ngl

es, a

nd

para

llel

ogra

ms.

5. S

KY

SCR

APE

RS

On

vac

atio

n, T

ony’

s fa

mil

y to

ok a

hel

icop

ter

tou

r of

th

e ci

ty.

Th

e pi

lot

said

th

e n

ewes

t bu

ildi

ng

in t

he

city

was

th

e bu

ildi

ng

wit

h

this

top

vie

w. H

e to

ld T

ony

that

th

e ex

teri

or a

ngl

e by

th

e fr

ont

entr

ance

is

72°

. Ton

y w

ante

d to

kn

ow

mor

e ab

out

the

buil

din

g, s

o h

e dr

ew t

his

di

agra

m a

nd

use

d h

is g

eom

etry

ski

lls

to

lear

n a

few

mor

e th

ings

. Th

e fr

ont

entr

ance

is

nex

t to

ver

tex

B.

a. W

hat

are

th

e m

easu

res

of t

he

fou

r an

gles

of

the

para

llel

ogra

m?

b.

How

man

y pa

irs

of c

ongr

uen

t tr

ian

gles

are

th

ere

in t

he

figu

re?

Wh

at a

re t

hey

?

Wor

d Pr

oble

m P

ract

ice

Para

llelo

gra

ms

AB

CD

E

72˚

para

llelo

gram

s

rect

angl

es

squa

res

Goal

ie B

Play

er A Go

alie

A

xPl

ayer

B

Tere

saTr

avis

Grac

ieKe

nny

a

e

6 -2

Y

es.

Op

po

sit

e s

ides o

f a

para

llelo

gra

m a

re p

ara

llel

an

d t

he

para

llel

pro

pert

y i

s t

ran

sit

ive.

3

mi

H

e o

r sh

e a

lso

has t

o s

ee

an

gle

x.

72,

72,

108,

108

4 p

air

s;

�A

BE

� �

DC

E,

�A

BC

� �

DC

B,

�A

CE

� �

DB

E,

an

d

�A

BD

� �

DC

A

001_

042_

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C06

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515.

indd

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:39:

27P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

16

G

lenc

oe G

eom

etry

Enri

chm

ent

Dia

go

nals

of

Para

llelo

gra

ms

In s

ome

draw

ings

th

e di

agon

al o

f a

para

llel

ogra

m a

ppea

rs t

o be

th

e an

gle

bise

ctor

of

both

op

posi

te a

ngl

es. W

hen

mig

ht

that

be

tru

e?

1. G

iven

: P

aral

lelo

gram

PQ

RS

wit

h d

iago

nal

−−

P

R .

−−

P

R i

s an

an

gle

bise

ctor

of

∠Q

PS

an

d ∠

QR

S.

W

hat

typ

e of

par

alle

logr

am i

s P

QR

S?

Just

ify

you

r an

swer

.

2. G

iven

: P

aral

lelo

gram

WP

RK

wit

h a

ngl

e

bise

ctor

−−

−

K

D , D

P =

5,

an

d W

D =

7.

F

ind

WK

an

d K

R.

3. R

efer

to

Exe

rcis

e 2.

Wri

te a

sta

tem

ent

abou

t pa

rall

elog

ram

WP

RK

an

d an

gle

bise

ctor

−−

−

K

D .

4. G

iven

: P

aral

lelo

gram

AB

CD

wit

h

diag

onal

−−

−

B

D a

nd

angl

e bi

sect

or −

−

B

P .

PD

=

5,

BP

=

6,

an

d C

P =

6.

T

he

peri

met

er o

f tr

ian

gle

PC

D i

s 15

.

Fin

d A

B a

nd

BC

.

P

QR

S

PD

A

BC

PD

RK

W

6 -2

∠

QR

S �

∠Q

PS

op

po

sit

e a

ng

les o

f a

para

llelo

gra

m

∠

QP

R �

∠R

PS

defi

nit

ion

of

an

gle

bis

ecto

r

∠

QR

P �

∠S

RP

defi

nit

ion

of

an

gle

bis

ecto

r

∠

QP

R �

∠P

RS

alt

ern

ate

in

teri

or

an

gle

s

∠

RP

S �

∠P

RS

Tra

nsit

ive P

rop

ert

y

−−

SP

� −

−

SR

Is

oscele

s T

rian

gle

Th

eo

rem

S

o a

ll s

ides a

re �

an

d Q

RP

S i

s a

rh

om

bu

s

W

K =

7,

KR

= 1

2

S

am

ple

an

sw

er:

Para

llelo

gra

m

WK

RP

is n

ot

a r

ho

mb

us a

nd

−−

KD

is

no

t a d

iag

on

al.

A

B =

4,

BC

= 9

001_

042_

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NA

ME

DA

TE

PE

RIO

D

Lesson 6-3

Ch

ap

ter

6

17

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

Tests

fo

r P

ara

llelo

gra

ms

Co

nd

itio

ns

for

Para

llelo

gra

ms

Th

ere

are

man

y w

ays

to

esta

blis

h t

hat

a q

uad

rila

tera

l is

a p

aral

lelo

gram

.

F

ind

x a

nd

y s

o th

at F

GH

J i

s a

par

alle

logr

am.

FG

HJ

is

a pa

rall

elog

ram

if

the

len

gth

s of

th

e op

posi

te

side

s ar

e eq

ual

. 6x

+ 3

= 1

5 4x

- 2

y =

2

6x =

12

4(2)

- 2

y =

2

x =

2

8 -

2y

= 2

-2y

= -

6

y

= 3

Exer

cise

sF

ind

x a

nd

y s

o th

at t

he

qu

adri

late

ral

is a

par

alle

logr

am.

1.

2.

3.

4.

5.

6.

AB

CD

E

JHG

F6x

+ 3

154x

- 2

y2

2x-

2

2y8

1211

x°

5y°

55°

5x°

5y°

25°

( x+

y)°

2x°

30°

24°

6y°

3x°

9x°

6y45

°18

6 -3

If:

If:

bo

th p

airs o

f o

pp

osite

sid

es a

re p

ara

llel,

−−

A

B |

| −

−−

D

C a

nd −

−

A

D |

| −

−−

B

C ,

bo

th p

airs o

f o

pp

osite

sid

es a

re c

on

gru

en

t, −

−

A

B �

−

−−

D

C a

nd

−

−

A

D �

−

−−

B

C ,

bo

th p

airs o

f o

pp

osite

an

gle

s a

re c

on

gru

en

t,∠

AB

C �

∠A

DC

and

∠D

AB

� ∠

BC

D,

the

dia

go

na

ls b

ise

ct

ea

ch

oth

er,

−−

A

E �

−−

C

E a

nd −

−

D

E �

−−

BE

,

on

e p

air o

f o

pp

osite

sid

es is c

on

gru

en

t a

nd

pa

ralle

l, −

−

A

B |

| −

−−

C

D a

nd −

−

A

B �

−−

−

C

D ,

or

−−

A

D |

| −

−−

B

C a

nd −

−

A

D �

−−

−

B

C ,

the

n:

the

fig

ure

is a

pa

ralle

log

ram

.th

en

: A

BC

D is a

pa

ralle

log

ram

.

Exam

ple

x =

7;

y =

4x =

5;

y =

25

x =

31;

y =

5x =

5;

y =

3

x =

15;

y =

9x =

30;

y =

15

001_

042_

GE

OC

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C06

_890

515.

indd

174/

11/0

811

:39:

38P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

18

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Tests

fo

r P

ara

llelo

gra

ms

Para

llelo

gra

ms

on

th

e C

oo

rdin

ate

Plan

e O

n t

he

coor

din

ate

plan

e, t

he

Dis

tan

ce,

Slo

pe, a

nd

Mid

poin

t F

orm

ula

s ca

n b

e u

sed

to t

est

if a

qu

adri

late

ral

is a

par

alle

logr

am.

D

eter

min

e w

het

her

AB

CD

is

a p

aral

lelo

gram

.

Th

e ve

rtic

es a

re A

(-2,

3),

B(3

, 2),

C(2

, -1)

, an

d D

(-3,

0).

Met

hod

1: U

se t

he

Slo

pe F

orm

ula

, m =

y 2 -

y1

−

x 2 -

x1 .

slop

e of

−−

−

A

D =

3 -

0

−

-2

- (

-3)

=

3 −

1 = 3

sl

ope

of −

−−

B

C =

2

- (

-1)

−

3 -

2

=

3 −

1 = 3

slop

e of

−−

A

B =

2 -

3

−

3 -

(-

2)

= -

1 −

5 sl

ope

of −

−−

C

D =

-1

- 0

−

2 -

(-

3)

= -

1 −

5

Sin

ce o

ppos

ite

side

s h

ave

the

sam

e sl

ope,

−−

A

B ||

−−

−

C

D a

nd

−−

−

A

D ||

−−

−

B

C . T

her

efor

e, A

BC

D i

s a

para

llel

ogra

m b

y de

fin

itio

n.

Met

hod

2: U

se t

he

Dis

tan

ce F

orm

ula

, d =

√ �

��

��

��

��

(x

2 -

x1)

2 +

( y

2 -

y1)

2 .

AB

= √

��

��

��

��

�

(-

2 -

3)2

+ (

3 -

2)2

= √

��

�

25 +

1 o

r √

��

26

CD

= √

��

��

��

��

��

(2

- (

-3)

)2 +

(-

1 -

0)2

= √

��

�

25 +

1 o

r √

��

26

AD

= √

��

��

��

��

��

(-

2 -

(-

3))2

+ (

3 -

0)2

= √

��

�

1 +

9 o

r √

��

10

BC

= √

��

��

��

��

�

(3

- 2

)2 +

(2

- (

-1)

)2 =

√ �

��

1 +

9 o

r √

��

10

Sin

ce b

oth

pai

rs o

f op

posi

te s

ides

hav

e th

e sa

me

len

gth

, −

−

A

B �

−−

−

C

D a

nd

−−

−

A

D �

−−

−

B

C . T

her

efor

e,

AB

CD

is

a pa

rall

elog

ram

by

Th

eore

m 6

.9.

Exer

cise

sG

rap

h e

ach

qu

adri

late

ral

wit

h t

he

give

n v

erti

ces.

Det

erm

ine

wh

eth

er t

he

figu

re i

s a

par

alle

logr

am. J

ust

ify

you

r an

swer

wit

h t

he

met

hod

in

dic

ated

.

1.

A(0

, 0),

B(1

, 3),

C(5

, 3),

D(4

, 0);

2.

D(-

1, 1

), E

(2, 4

), F

(6, 4

), G

(3, 1

);S

lope

For

mu

la

Slo

pe F

orm

ula

3.

R(-

1, 0

), S

(3, 0

), T

(2, -

3), U

(-3,

-2)

; 4.

A(-

3, 2

), B

(-1,

4),

C(2

, 1),

D(0

, -1)

;D

ista

nce

For

mu

la

Dis

tan

ce a

nd

Slo

pe F

orm

ula

s

5.

S(-

2, 4

), T

(-1,

-1)

, U(3

, -4)

, V(2

, 1);

6.

F(3

, 3),

G(1

, 2),

H(-

3, 1

), I

(-1,

4);

Dis

tan

ce a

nd

Slo

pe F

orm

ula

s M

idpo

int

For

mu

la

7. A

par

alle

logr

am h

as v

erti

ces

R(-

2, -

1), S

(2, 1

), a

nd

T(0

, -3)

. Fin

d al

l po

ssib

le

coor

din

ates

for

th

e fo

urt

h v

erte

x.

x

y

O

C

A

D

B

6 -3

Exam

ple

y

es

yes

n

o

yes

y

es

n

o

(

4,

-1),

(0,

3),

or

(-4,

-5)

S

ee s

tud

en

ts’

wo

rk

001_

042_

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OC

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indd

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M


NA

ME

DA

TE

PE

RIO

D

Lesson 6-3

Ch

ap

ter

6

19

G

lenc

oe G

eom

etry

Skill

s Pr

acti

ceTests

fo

r P

ara

llelo

gra

ms

Det

erm

ine

wh

eth

er e

ach

qu

adri

late

ral

is a

par

alle

logr

am. J

ust

ify

you

r an

swer

.

1.

2.

3.

4.

CO

OR

DIN

ATE

GEO

MET

RY

Gra

ph

eac

h q

uad

rila

tera

l w

ith

th

e gi

ven

ver

tice

s.

Det

erm

ine

wh

eth

er t

he

figu

re i

s a

par

alle

logr

am. J

ust

ify

you

r an

swer

wit

h t

he

met

hod

in

dic

ated

.

5. P

(0, 0

), Q

(3, 4

), S

(7, 4

), Y

(4, 0

); S

lope

For

mu

la

6. S

(-2,

1),

R(1

, 3),

T(2

, 0),

Z(-

1, -

2); D

ista

nce

an

d S

lope

For

mu

las

7. W

(2, 5

), R

(3, 3

), Y

(-2,

-3)

, N(-

3, 1

); M

idpo

int

For

mu

la

ALG

EBR

A F

ind

x a

nd

y s

o th

at e

ach

qu

adri

late

ral

is a

par

alle

logr

am.

8.

9.

10.

11.

x+

16

y+

19

2y

2x-

82y

- 1

1

y+

3

4x - 3

3x

( 4x

- 3

5)°

( 3x

+ 1

0)°

( 2y

- 5

) °

( y+

15)

°

3x-

14

x+

20

3y+

2y

+ 2

0

6 -3

Y

es;

a p

air

of

op

po

sit

e s

ides

Yes;

bo

th p

air

s o

f o

pp

osit

eis

para

llel

an

d c

on

gru

en

t.

an

gle

s a

re c

on

gru

en

t.

N

o;

no

ne o

f th

e t

ests

fo

r

Y

es;

bo

th p

air

s o

f o

pp

osit

e s

ides

para

llelo

gra

ms i

s f

ulf

ille

d.

are

co

ng

ruen

t.

Y

es;

SR

= Z

T a

nd

th

e s

lop

es o

f −

−

SR

an

d −

−

ZT a

re e

qu

al, s

o o

ne p

air

of

op

po

sit

e s

ides i

s p

ara

llel

an

d c

og

ruen

t.

x

= 2

4,

y =

19

x =

3,

y =

14

x

= 4

5,

y =

20

x =

17,

y =

9

N

o;

the m

idp

oin

ts o

f th

e d

iag

on

als

are

no

t th

e s

am

e p

oin

t.

Y

es;

the s

lop

e o

f −

−

PY

an

d −

−

QS

are

eq

ual

an

d t

he s

lop

e o

f −

−

PQ

an

d −

−

YS

are

eq

ual, s

o −

−

PY

‖ −

−

QS

an

d −

−

PQ

‖ −

−

YS

. O

pp

osit

e s

ides a

re p

ara

llel.

S

ee s

tud

en

ts’

gra

ph

s.

001_

042_

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C06

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49:4

9P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

20

G

lenc

oe G

eom

etry

Det

erm

ine

wh

eth

er e

ach

qu

adri

late

ral

is a

par

alle

logr

am. J

ust

ify

you

r an

swer

.

1.

2.

3.

4.

CO

OR

DIN

ATE

GEO

MET

RY

Gra

ph

eac

h q

uad

rila

tera

l w

ith

th

e gi

ven

ver

tice

s.

Det

erm

ine

wh

eth

er t

he

figu

re i

s a

par

alle

logr

am. J

ust

ify

you

r an

swer

wit

h t

he

met

hod

in

dic

ated

.

5. P

(-5,

1),

S(-

2, 2

), F

(-1,

-3)

, T(2

, -2)

; Slo

pe F

orm

ula

6. R

(-2,

5),

O(1

, 3),

M(-

3, -

4), Y

(-6,

-2)

; Dis

tan

ce a

nd

Slo

pe F

orm

ula

s

ALG

EBR

A F

ind

x a

nd

y s

o th

at t

he

qu

adri

late

ral

is a

par

alle

logr

am.

7.

8.

9.

10.

11. T

ILE

DES

IGN

Th

e pa

tter

n s

how

n i

n t

he

figu

re i

s to

con

sist

of

con

gru

ent

para

llel

ogra

ms.

How

can

th

e de

sign

er b

e ce

rtai

n t

hat

th

e sh

apes

are

pa

rall

elog

ram

s?

118°

62°

118°

62°

( 5x

+ 2

9)°

( 7x

- 1

1)°

( 3y

+ 1

5)°

( 5y

- 9

) °

3y-

5-

3x+

4

-4x

- 2

2y+

8

-4x

+ 6

-6x

7y+

312

y-

7-

4y-

2

x + 12

-2x

+ 6

y+

23

Prac

tice

Tests

fo

r P

ara

llelo

gra

ms

6 -3

Y

es;

the d

iag

on

als

bis

ect

No

; n

on

e o

f th

e t

ests

fo

r

each

oth

er.

para

llelo

gra

ms i

s f

ulf

ille

d.

Y

es;

bo

th p

air

s o

f o

pp

osit

e

No

; n

on

e o

f th

e t

ests

fo

ran

gle

s a

re c

on

gru

en

t.

p

ara

llelo

gra

ms i

s f

ulf

ille

d.

y

es

x

= 2

0,

y =

12

x =

-6,

y =

13

x

= -

3,

y =

2

x =

-2,

y =

-5

S

am

ple

an

sw

er:

Co

nfi

rm t

hat

bo

th p

air

s o

f o

pp

osit

e �

are

�.

y

es

S

ee s

tud

en

ts’

wo

rk

001_

042_

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OC

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C06

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54P

M

Lesson 6-3


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

21

G

lenc

oe G

eom

etry

1. B

ALA

NC

ING

Nik

ia, M

adis

on, A

nge

la,

and

Sh

elby

are

bal

anci

ng

them

selv

es

on a

n “

X”-

shap

ed f

loat

ing

obje

ct. T

o ba

lan

ce t

hem

selv

es, t

hey

wan

t to

m

ake

them

selv

es t

he

vert

ices

of

a pa

rall

elog

ram

.

In o

rder

to

ach

ieve

th

is, d

o al

l fo

ur

of

them

hav

e to

be

the

sam

e di

stan

ce f

rom

th

e ce

nte

r of

th

e ob

ject

? E

xpla

in.

2. C

OM

PASS

ES T

wo

com

pass

nee

dles

pl

aced

sid

e by

sid

e on

a t

able

are

bot

h

2 in

ches

lon

g an

d po

int

due

nor

th. D

o th

ey f

orm

th

e si

des

of a

par

alle

logr

am?

3. F

OR

MA

TIO

N F

our

jets

are

fly

ing

in

form

atio

n. T

hre

e of

th

e je

ts a

re s

how

n

in t

he

grap

h. I

f th

e fo

ur

jets

are

loc

ated

at

th

e ve

rtic

es o

f a

para

llel

ogra

m, w

hat

ar

e th

e th

ree

poss

ible

loc

atio

ns

of t

he

mis

sin

g je

t?

4. S

TREE

T LA

MPS

Wh

en a

coo

rdin

ate

plan

e is

pla

ced

over

th

e H

arri

svil

le t

own

m

ap, t

he

fou

r st

reet

lam

ps i

n t

he

cen

ter

are

loca

ted

as s

how

n. D

o th

e fo

ur

lam

ps

form

th

e ve

rtic

es o

f a

para

llel

ogra

m?

Exp

lain

.

5. P

ICTU

RE

FRA

ME

Aar

on i

s m

akin

g a

woo

den

pic

ture

fra

me

in t

he

shap

e of

a

para

llel

ogra

m. H

e h

as t

wo

piec

es o

f w

ood

that

are

3 f

eet

lon

g an

d tw

o th

at

are

4 fe

et l

ong.

a. I

f h

e co

nn

ects

th

e pi

eces

of

woo

d at

th

eir

ends

to

each

oth

er, i

n w

hat

or

der

mu

st h

e co

nn

ect

them

to

mak

e a

para

llel

ogra

m?

b.

How

man

y di

ffer

ent

para

llel

ogra

ms

cou

ld h

e m

ake

wit

h t

hes

e fo

ur

len

gth

s of

woo

d?

c. E

xpla

in s

omet

hin

g A

aron

mig

ht

do

to s

peci

fy p

reci

sely

th

e sh

ape

of t

he

para

llel

ogra

m.

5

5y

xO

–5

–5

5

5

y

xO

Shel

by

Mad

ison

Niki

aAn

gela

6 -3

Wor

d Pr

oble

m P

ract

ice

Tests

fo

r P

ara

llelo

gra

ms

N

o,

Mad

iso

n a

nd

An

gela

have t

o

be t

he s

am

e d

ista

nce a

nd

Nik

ia

an

d S

helb

y h

ave t

o b

e t

he s

am

e

dis

tan

ce b

ut

Nik

ia a

nd

Sh

elb

y’s

d

ista

nce f

rom

th

e c

en

ter

do

es

no

t h

ave t

o b

e e

qu

al

to M

ad

iso

n

an

d A

ng

ela

’s d

ista

nce.

y

es

(

1,

7),

(9,

-1),

(-

7,

-3)

Y

es;

the l

en

gth

s o

f th

e o

pp

osit

e

sid

es a

re c

on

gru

en

t.

He m

ust

alt

ern

ate

th

e l

en

gth

s

3,

4,

3,

4 o

r 4,

3,

4,

3.

infi

nit

ely

man

y

Sam

ple

an

sw

er:

He c

ou

ld

sp

ecif

y t

he l

en

gth

of

a

dia

go

nal.

001_

042_

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00P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

22

G

lenc

oe G

eom

etry

Enri

chm

ent

Tests

fo

r P

ara

llelo

gra

ms

By

defi

nit

ion

, a q

uad

rila

tera

l is

a p

aral

lelo

gram

if

and

on

ly i

f bo

th p

airs

of

oppo

site

sid

es

are

para

llel

. Wh

at c

ondi

tion

s ot

her

th

an b

oth

pai

rs o

f op

posi

te s

ides

par

alle

l w

ill

guar

ante

e th

at a

qu

adri

late

ral

is a

par

alle

logr

am?

In t

his

act

ivit

y, s

ever

al p

ossi

bili

ties

wil

l be

in

vest

igat

ed b

y dr

awin

g qu

adri

late

rals

to

sati

sfy

cert

ain

con

diti

ons.

Rem

embe

r th

at a

ny

test

th

at s

eem

s to

wor

k is

not

gu

aran

teed

to

wor

k u

nle

ss i

t ca

n b

e fo

rmal

ly p

rove

n.

Com

ple

te.

1. D

raw

a q

uad

rila

tera

l w

ith

on

e pa

ir o

f op

posi

te s

ides

con

gru

ent.

M

ust

it

be a

par

alle

logr

am?

2. D

raw

a q

uad

rila

tera

l w

ith

bot

h p

airs

of

oppo

site

sid

es c

ongr

uen

t.

Mu

st i

t be

a p

aral

lelo

gram

?

3. D

raw

a q

uad

rila

tera

l w

ith

on

e pa

ir o

f op

posi

te s

ides

par

alle

l an

d

the

oth

er p

air

of o

ppos

ite

side

s co

ngr

uen

t. M

ust

it

be a

pa

rall

elog

ram

?

4. D

raw

a q

uad

rila

tera

l w

ith

on

e pa

ir o

f op

posi

te s

ides

bot

h p

aral

lel

an

d co

ngr

uen

t. M

ust

it

be a

par

alle

logr

am?

5. D

raw

a q

uad

rila

tera

l w

ith

on

e pa

ir o

f op

posi

te a

ngl

es c

ongr

uen

t.

Mu

st i

t be

a p

aral

lelo

gram

?

6. D

raw

a q

uad

rila

tera

l w

ith

bot

h p

airs

of

oppo

site

an

gles

con

gru

ent.

M

ust

it

be a

par

alle

logr

am?

7. D

raw

a q

uad

rila

tera

l w

ith

on

e pa

ir o

f op

posi

te s

ides

par

alle

l an

d

one

pair

of

oppo

site

an

gles

con

gru

ent.

Mu

st i

t be

a p

aral

lelo

gram

?

6 -3

no

yes

no

yes

no

yes

yes

001_

042_

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M

Lesson 6-4


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

23

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

Recta

ng

les

Pro

per

ties

of

Rec

tan

gle

s A

rec

tan

gle

is a

qu

adri

late

ral

wit

h f

our

ri

ght

angl

es. H

ere

are

the

prop

erti

es o

f re

ctan

gles

.A

rec

tan

gle

has

all

th

e pr

oper

ties

of

a pa

rall

elog

ram

.•

Opp

osit

e si

des

are

para

llel

.•

Opp

osit

e an

gles

are

con

gru

ent.

• O

ppos

ite

side

s ar

e co

ngr

uen

t.•

Con

secu

tive

an

gles

are

su

pple

men

tary

.•

Th

e di

agon

als

bise

ct e

ach

oth

er.

Als

o:•

All

fou

r an

gles

are

rig

ht

angl

es.

∠U

TS

, ∠T

SR

, ∠S

RU

, an

d ∠

RU

T a

re r

igh

t an

gles

.•

Th

e di

agon

als

are

con

gru

ent.

−

−

T

R �

−−

−

U

S

TS R

U

Q

Q

uad

rila

tera

l R

UT

S

abov

e is

a r

ecta

ngl

e. I

f U

S =

6x

+ 3

an

d

RT

= 7

x -

2, f

ind

x.

Th

e di

agon

als

of a

rec

tan

gle

are

con

gru

ent,

so

US

= R

T.

6x +

3 =

7x

- 2

3

= x

- 2

5

= x

Q

uad

rila

tera

l R

UT

S

abov

e is

a r

ecta

ngl

e. I

f m

∠S

TR

= 8

x +

3

and

m∠

UT

R =

16x

- 9

, fin

d m

∠S

TR

.

∠U

TS

is

a ri

ght

angl

e, s

o m

∠S

TR

+ m

∠U

TR

= 9

0. 8x

+ 3

+ 1

6x -

9 =

90

24

x -

6 =

90

24

x =

96

x

= 4

m∠

ST

R =

8x

+ 3

= 8

(4)

+ 3

or

35

Exer

cise

sQ

uad

rila

tera

l A

BC

D i

s a

rect

angl

e.

1. I

f A

E =

36

and

CE

= 2

x -

4, f

ind

x.

2. I

f B

E =

6y

+ 2

an

d C

E =

4y

+ 6

, fin

d y.

3. I

f B

C =

24

and

AD

= 5

y -

1, f

ind

y.

4. I

f m

∠B

EA

= 6

2, f

ind

m∠

BA

C.

5. I

f m

∠A

ED

= 1

2x a

nd

m∠

BE

C =

10x

+ 2

0, f

ind

m∠

AE

D.

6. I

f B

D =

8y

- 4

an

d A

C =

7y

+ 3

, fin

d B

D.

7. I

f m

∠D

BC

= 1

0x a

nd

m∠

AC

B =

4x2 -

6, f

ind

m∠

AC

B.

8. I

f A

B =

6y

and

BC

= 8

y, f

ind

BD

in

ter

ms

of y

.

BC D

A

E

6 -4

Exam

ple

1Ex

amp

le 2

20

2

5

59

120

52

30

10

y

001_

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234/

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10P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

24

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Recta

ng

les

Pro

ve t

hat

Par

alle

log

ram

s A

re R

ecta

ng

les

Th

e di

agon

als

of a

rec

tan

gle

are

con

gru

ent,

an

d th

e co

nve

rse

is a

lso

tru

e.

If t

he

dia

go

na

ls o

f a

pa

ralle

log

ram

are

co

ng

rue

nt,

th

en

th

e p

ara

llelo

gra

m is a

re

cta

ng

le.

In t

he

coor

din

ate

plan

e yo

u c

an u

se t

he

Dis

tan

ce F

orm

ula

, th

e S

lope

For

mu

la, a

nd

prop

erti

es o

f di

agon

als

to s

how

th

at a

fig

ure

is

a re

ctan

gle.

Q

uad

rila

tera

l A

BC

D h

as v

erti

ces

A(-

3, 0

), B

(-2,

3),

C

(4, 1

), a

nd

D(3

, -2)

. Det

erm

ine

wh

eth

er A

BC

D i

s a

rect

angl

e.

Met

hod

1: U

se t

he

Slo

pe F

orm

ula

.

slop

e of

−−

A

B =

3 -

0

−

-2

- (-

3)

= 3 −

1 or

3 sl

ope

of −

−−

A

D =

-2

- 0

−

3 -

(-

3)

= -

2 −

6 or

- 1 −

3

slop

e of

−−

−

C

D =

-

2 -

1

−

3 -

4

= -

3 −

-1 o

r 3

slop

e of

−−

−

B

C =

1 -

3

−

4 -

(-

2)

= -

2 −

6 or

- 1 −

3

Opp

osit

e si

des

are

para

llel

, so

the

figu

re i

s a

para

llel

ogra

m. C

onse

cuti

ve s

ides

are

pe

rpen

dicu

lar,

so

AB

CD

is

a re

ctan

gle.

x

y O

D

B

A

EC

6 -4

Exam

ple

Met

hod

2: U

se t

he

Dis

tan

ce F

orm

ula

.

AB

= √

��

��

��

��

��

(-

3 -

(-

2) ) 2

+ (

0 -

3 ) 2

or

√ �

�

10

BC

= √

��

��

��

��

(-

2 -

4 ) 2

+ (

3 -

1 ) 2

or

√ �

�

40

CD

= √

��

��

��

��

�

(4

- 3

) 2 +

(1

- (

-2)

) 2 o

r √

��

10

AD

= √

��

��

��

��

��

(-3

- 3

) 2 +

(0

- (

-2)

) 2 o

r √

��

40

Opp

osit

e si

des

are

con

gru

ent,

th

us

AB

CD

is

a pa

rall

elog

ram

.

AC

= √

��

��

��

��

(-

3 -

4 ) 2

+ (

0 -

1 ) 2

or

√ �

�

50

BD

= √

��

��

��

��

��

(-2

- 3

) 2 +

(3

- (

-2)

) 2 o

r √

��

50

AB

CD

is

a pa

rall

elog

ram

wit

h c

ongr

uen

t di

agon

als,

so

AB

CD

is

a re

ctan

gle.

Exer

cise

sC

OO

RD

INA

TE G

EOM

ETR

Y G

rap

h e

ach

qu

adri

late

ral

wit

h t

he

give

n v

erti

ces.

D

eter

min

e w

het

her

th

e fi

gure

is

a re

ctan

gle.

Ju

stif

y yo

ur

answ

er u

sin

g th

e in

dic

ated

for

mu

la.

1. A

(−3,

1),

B(−

3, 3

), C

(3, 3

), D

(3, 1

); D

ista

nce

For

mu

la

2. A

(−3,

0),

B(−

2, 3

), C

(4, 5

), D

(3, 2

); S

lope

For

mu

la

3. A

(−3,

0),

B(−

2, 2

), C

(3, 0

), D

(2 −

2); D

ista

nce

For

mu

la

4. A

(−1,

0),

B(0

, 2),

C(4

, 0),

D(3

, −2)

; Dis

tan

ce F

orm

ula

Y

es;

AB

= 2

, B

C =

6,

CD

= 2

, D

A =

6,

AC

= √

��

40 ,

BD

= √

��

40 ,

op

po

sit

e

sid

es a

nd

dia

go

nals

are

co

ng

ruen

t.

N

o;

slo

pe o

f −

−

AB

= 3

, slo

pe o

f −

−

BC

= 1

−

3 ,

slo

pes s

ho

w t

hat

two

co

nsecu

tive

sid

es a

re n

ot

perp

en

dic

ula

r.

S

ee s

tud

en

ts’

wo

rk

N

o;

AC

= 6

, B

D =

√ �

�

32 ,

dia

go

nals

are

no

t co

ng

ruen

t.

CD

= √

�

5 ,

DA

= √

��

20 ,

AC

= 5

, B

D =

5,

op

po

sit

e s

ides a

nd

dia

go

nals

are

co

ng

ruen

t.

Y

es;

AB

= √

�

5 ,

BC

= √

��

20 ,

001_

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11/0

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16P

M

Lesson 6-4


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

25

G

lenc

oe G

eom

etry

Skill

s Pr

acti

ceR

ecta

ng

les

ALG

EBR

A Q

uad

rila

tera

l A

BC

D i

s a

rect

angl

e.

1. I

f A

C =

2x

+ 1

3 an

d D

B =

4x

- 1

, fin

d D

B.

2. I

f A

C =

x +

3 a

nd

DB

= 3

x -

19,

fin

d A

C.

3. I

f A

E =

3x

+ 3

an

d E

C =

5x

- 1

5, f

ind

AC

.

4. I

f D

E =

6x

- 7

an

d A

E =

4x

+ 9

, fin

d D

B.

5. I

f m

∠D

AC

= 2

x +

4 a

nd

m∠

BA

C =

3x

+ 1

, fin

d m

∠B

AC

.

6. I

f m

∠B

DC

= 7

x +

1 a

nd

m∠

AD

B =

9x

- 7

, fin

d m

∠B

DC

.

7. I

f m

∠A

BD

= 7

x -

31

and

m∠

CD

B =

4x

+ 5

, fin

d m

∠A

BD

.

8. I

f m

∠B

AC

= x

+ 3

an

d m

∠C

AD

= x

+ 1

5, f

ind

m∠

BA

C.

9. P

RO

OF:

Wri

te a

tw

o-co

lum

n p

roof

.

G

iven

: RS

TV

is

a re

ctan

gle

and

U i

s th

e m

idpo

int

of −

−

V

T .

P

rove

: �R

UV

� �

SU

T

Pro

of

S

tate

men

ts

Rea

son

s

1. R

STV

is a

recta

ng

le.

1.

Giv

en

2.

∠V

an

d ∠

T a

re r

igh

t an

gle

s.

2.

Defi

nit

ion

of

recta

ng

le

3.

∠V

� ∠

T

3.

All r

t �

are

�.

4.

U i

s t

he m

idp

oin

t o

f −

−

VT .

4.

Giv

en

5.

−−

VU

� −

−

TU

5.

Defi

nit

ion

of

mid

po

int

6.

−−

VR

� −

−

TS

6.

Op

p s

ides o

f �

are

co

ng

ruen

t.

7.

R

UV

�

SU

T

7.

SA

S

CO

OR

DIN

ATE

GEO

MET

RY

Gra

ph

eac

h q

uad

rila

tera

l w

ith

th

e gi

ven

ver

tice

s.

Det

erm

ine

wh

eth

er t

he

figu

re i

s a

rect

angl

e. J

ust

ify

you

r an

swer

usi

ng

the

ind

icat

ed f

orm

ula

.

10. P

(-3,

-2)

, Q(-

4, 2

), R

(2, 4

), S

(3, 0

); S

lope

For

mu

la

11. J

(-6,

3),

K(0

, 6),

L(2

, 2),

M(-

4, -

1); D

ista

nce

For

mu

la

12. T

(4, 1

), U

(3, -

1), X

(-3,

2),

Y(-

2, 4

); D

ista

nce

For

mu

la

DCB

AE

6-4

27

14 60

82

52

43 5

3

39

N

o;

Sam

ple

an

sw

er:

An

gle

s a

re n

ot

rig

ht

an

gle

s.

Y

es;

Sam

ple

an

sw

er:

Bo

th p

air

s o

f o

pp

osit

e s

ides a

re c

on

gru

en

t

an

d d

iag

on

als

are

co

ng

ruen

t.

Y

es;

Sam

ple

an

sw

er:

Bo

th p

air

s o

f o

pp

osit

e s

ides a

re c

on

gru

en

t an

d t

he

dia

go

nals

are

co

ng

ruen

t.

S

ee s

tud

en

ts’

gra

ph

s.

001_

042_

GE

OC

RM

C06

_890

515.

indd

255/

28/0

82:

28:1

2P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

26

G

lenc

oe G

eom

etry

Prac

tice

Recta

ng

les

ALG

EBR

A Q

uad

rila

tera

l R

ST

U i

s a

rect

angl

e.

1. I

f U

Z=

x+

21

and

ZS

= 3

x-

15,

fin

d U

S.

2. I

f R

Z =

3x

+ 8

an

d Z

S =

6x

- 2

8, f

ind

UZ

.

3. I

f R

T =

5x

+ 8

an

d R

Z =

4x

+ 1

, fin

d Z

T.

4. I

f m

∠S

UT

= 3

x +

6 a

nd

m∠

RU

S =

5x

- 4

, fin

d m

∠S

UT

.

5. I

f m

∠S

RT

= x

+ 9

an

d m

∠U

TR

= 2

x -

44,

fin

d m

∠U

TR

.

6. I

f m

∠R

SU

= x

+ 4

1 an

d m

∠T

US

= 3

x +

9, f

ind

m∠

RS

U.

Qu

adri

late

ral

GH

JK

is

a re

ctan

gle.

Fin

d e

ach

mea

sure

if

m∠

1 =

37.

7. m

∠2

8. m

∠3

9. m

∠4

10. m

∠5

11. m

∠6

12. m

∠7

CO

OR

DIN

ATE

GEO

MET

RY

Gra

ph

eac

h q

uad

rila

tera

l w

ith

th

e gi

ven

ver

tice

s.

Det

erm

ine

wh

eth

er t

he

figu

re i

s a

rect

angl

e. J

ust

ify

you

r an

swer

usi

ng

the

ind

icat

ed f

orm

ula

.

13. B

(-4,

3),

G(-

2, 4

), H

(1, -

2), L

(-1,

-3)

; Slo

pe F

orm

ula

14. N

(-4,

5),

O(6

, 0),

P(3

, -6)

, Q(-

7, -

1); D

ista

nce

For

mu

la

15. C

(0, 5

), D

(4, 7

), E

(5, 4

), F

(1, 2

); S

lope

For

mu

la

16. L

AN

DSC

API

NG

Hu

nti

ngt

on P

ark

offi

cial

s ap

prov

ed a

rec

tan

gula

r pl

ot o

f la

nd

for

a Ja

pan

ese

Zen

gar

den

. Is

it s

uff

icie

nt

to k

now

th

at o

ppos

ite

side

s of

th

e ga

rden

plo

t ar

e co

ngr

uen

t an

d pa

rall

el t

o de

term

ine

that

th

e ga

rden

plo

t is

rec

tan

gula

r? E

xpla

in.

UTS

RZ

K

21

5

43

67

JHG

6 -4

78

44

9

39

62

57

53

37

37

53

106

74

Y

es;

sam

ple

an

sw

er:

Op

po

sit

e s

ides a

re p

ara

llel

an

d c

on

secu

tive s

ides

are

perp

en

dic

ula

r.

Y

es;

sam

ple

an

sw

er:

Op

po

sit

e s

ides a

re c

on

gru

en

t an

d d

iag

on

als

are

co

ng

ruen

t.

N

o;

sam

ple

an

sw

er:

Dia

go

nals

are

no

t co

ng

ruen

t.

N

o;

if y

ou

on

ly k

no

w t

hat

op

po

sit

e s

ides a

re c

on

gru

en

t an

d p

ara

llel, t

he

mo

st

yo

u c

an

co

nclu

de i

s t

hat

the p

lot

is a

para

llelo

gra

m.

S

ee s

tud

en

ts’

wo

rk

001_

042_

GE

OC

RM

C06

_890

515.

indd

264/

12/0

85:

42:5

8A

M

Lesson 6-4


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

27

G

lenc

oe G

eom

etry

1. F

RA

MES

Jal

en m

akes

th

e re

ctan

gula

r fr

ame

show

n.

In o

rder

to

mak

e su

re t

hat

it

is a

re

ctan

gle,

Jal

en m

easu

res

the

dist

ance

s B

D a

nd

AC

. How

sh

ould

th

ese

two

dist

ance

s co

mpa

re i

f th

e fr

ame

is a

re

ctan

gle?

2. B

OO

KSH

ELV

ES A

bo

oksh

elf

con

sist

s of

tw

o ve

rtic

al p

lan

ks

wit

h f

ive

hor

izon

tal

shel

ves.

Are

eac

h o

f th

e fo

ur

sect

ion

s fo

r bo

oks

rect

angl

es?

Exp

lain

.

3. L

AN

DSC

API

NG

A l

ands

cape

r is

m

arki

ng

off

the

corn

ers

of a

rec

tan

gula

r pl

ot o

f la

nd.

Th

ree

of t

he

corn

ers

are

in

plac

e as

sh

own

.

Wh

at a

re t

he

coor

din

ates

of

the

fou

rth

co

rner

?

4. S

WIM

MIN

G P

OO

LS A

nto

nio

is

desi

gnin

g a

swim

min

g po

ol o

n a

co

ordi

nat

e gr

id. I

s it

a r

ecta

ngl

e?

Exp

lain

.

5. P

ATT

ERN

SV

eron

ica

mad

e th

e pa

tter

n

show

n o

ut

of 7

rec

tan

gles

wit

h f

our

equ

al s

ides

. Th

e si

de l

engt

h o

f ea

ch

rect

angl

e is

wri

tten

in

side

th

e re

ctan

gle.

a. H

ow m

any

rect

angl

es c

an b

e fo

rmed

u

sin

g th

e li

nes

in

th

is f

igu

re?

b.

If V

eron

ica

wan

ted

to e

xten

d h

er

patt

ern

by

addi

ng

anot

her

rec

tan

gle

wit

h 4

equ

al s

ides

to

mak

e a

larg

er

rect

angl

e, w

hat

are

th

e po

ssib

le

side

len

gth

s of

rec

tan

gles

th

at s

he

can

add

?

Wor

d Pr

oble

m P

ract

ice

Recta

ng

les

1 1 2

3

5

85y

xO

5

5y

xO

–5

–5

AB

DC

6 -4

T

hey s

ho

uld

be e

qu

al.

Y

es;

each

of

the f

ou

r an

gle

s o

f

each

recta

ng

le i

s c

reate

d b

y a

str

aig

ht

ho

rizo

nta

l lin

e a

nd

a

str

aig

ht

vert

ical

lin

e,

so

each

h

as f

ou

r 90

° an

gle

s.

(-6,

3)

Y

es.

Sam

ple

an

sw

er:

Th

e s

lop

e o

f th

e l

on

g s

ides i

s 3

an

d t

he s

lop

e

o

f th

e s

ho

rt s

ides i

s -

1

−

3 ,

so

each

p

air

of

sid

es i

s p

ara

llel. A

lso

, th

e

lon

g a

nd

sh

ort

sid

es h

ave s

lop

es

that

are

perp

en

dic

ula

r to

each

oth

er,

makin

g t

he f

ou

r co

rners

ri

gh

t an

gle

s.

11

8 a

nd

13 u

nit

s

001_

042_

GE

OC

RM

C06

_890

515.

indd

274/

11/0

811

:40:

33P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

28

G

lenc

oe G

eom

etry

Co

nsta

nt

Peri

mete

rD

ougl

as w

ants

to

fen

ce a

rec

tan

gula

r re

gion

of

his

bac

k ya

rd f

or h

is d

og. H

e bo

ugh

t 20

0 fe

et o

f fe

nce

.

1. C

ompl

ete

the

tabl

e to

sh

ow t

he

dim

ensi

ons

of f

ive

diff

eren

t re

ctan

gula

r pe

ns

that

wou

ld u

se t

he

enti

re 2

00 f

eet

of f

ence

. Th

en f

ind

the

area

of

each

re

ctan

gula

r pe

n.

2. D

o al

l fi

ve o

f th

e re

ctan

gula

r pe

ns

hav

e th

e sa

me

area

? If

not

, wh

ich

on

e h

as

the

larg

er a

rea?

3. W

rite

a r

ule

for

fin

din

g th

e di

men

sion

s of

a r

ecta

ngl

e w

ith

th

e la

rges

t po

ssib

le a

rea

for

a gi

ven

per

imet

er.

4. L

et x

rep

rese

nt

the

len

gth

of

a re

ctan

gle

and

y th

e w

idth

. W

rite

th

e fo

rmu

la f

or a

ll r

ecta

ngl

es w

ith

a p

erim

eter

of

200.

Th

en g

raph

th

is r

elat

ion

ship

on

th

e co

ordi

nat

e pl

ane

at t

he

righ

t.

Juli

o re

ad t

hat

a d

og t

he

size

of

his

new

pet

, Ben

nie

, sh

ould

hav

e at

lea

st 1

00 s

quar

e fe

et i

n

his

pen

. Bef

ore

goin

g to

th

e st

ore

to b

uy

fen

ce, J

uli

o m

ade

a ta

ble

to d

eter

min

e th

e di

men

sion

s fo

r B

enn

ie’s

rec

tan

gula

r pe

n.

5. C

ompl

ete

the

tabl

e to

fin

d fi

ve p

ossi

ble

dim

ensi

ons

of a

rec

tan

gula

r fe

nce

d ar

ea o

f 10

0 sq

uar

e fe

et.

6. J

uli

o w

ants

to

save

mon

ey b

y pu

rch

asin

g th

e le

ast

nu

mbe

r fe

et o

f fe

nci

ng

to e

ncl

ose

the

100

squ

are

feet

. Wh

at w

ill

be t

he

dim

ensi

ons

of t

he

com

plet

ed p

en?

7. W

rite

a r

ule

for

fin

din

g th

e di

men

sion

s of

a r

ecta

ngl

e w

ith

th

e le

ast

poss

ible

per

imet

er f

or a

giv

en a

rea.

8. F

or l

engt

h x

an

d w

idth

y, w

rite

a f

orm

ula

for

th

e ar

ea o

f a

rect

angl

e w

ith

an

are

a of

100

squ

are

feet

. Th

en g

raph

th

e fo

rmu

la.

100

100

y

xOy

xO

100

100

6-4

Pe

rim

ete

rL

en

gth

Wid

thA

rea

20

08

0

20

07

0

20

06

0

20

05

0

20

04

5

Are

aL

en

gth

Wid

thH

ow

mu

ch

fen

ce

to

bu

y

10

0

10

0

10

0

10

0

10

0

Enri

chm

ent

N

o,

the 5

0 ×

50 p

en

has t

he l

arg

est

are

a.

T

he r

ecta

ng

le w

ith

th

e l

arg

est

are

a f

or

a g

iven

peri

mete

r is

a s

qu

are

.

2

x +

2y =

200 o

r x +

y =

100

S

ee t

ab

le f

or

sam

ple

an

sw

ers

.

1

0 f

t ×

10 f

t

T

he r

ecta

ng

le w

ith

th

e l

east

po

ssib

le p

eri

mete

r fo

r a g

iven

are

a i

s a

sq

uare

.

x

y=

10

0

11

00

202

250

104

425

58

520

50

10

10

40

20

1600

30

2100

40

2400

50

2500

55

2475

001_

042_

GE

OC

RM

C06

_890

515.

indd

284/

11/0

811

:40:

39P

M

Lesson 6-4


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

29

G

lenc

oe G

eom

etry

Gra

phin

g Ca

lcul

ator

Act

ivit

yTI-

Nsp

ire:

Exp

lori

ng

Recta

ng

les

A q

uad

rila

tera

l w

ith

fou

r ri

ght

angl

es i

s a

rect

angl

e. T

he

TI-

Nsp

ire

can

be

use

d to

exp

lore

so

me

of t

he

char

acte

rist

ics

of a

rec

tan

gle.

Use

th

e fo

llow

ing

step

s to

dra

w a

rec

tan

gle.

Ste

p 1

S

et u

p th

e ca

lcu

lato

r in

th

e co

rrec

t m

ode.

• C

hoo

se G

rap

hs

& G

eom

etry

fro

m t

he

Hom

e M

enu

.•

Fro

m t

he

Vie

w m

enu

, ch

oose

4: H

ide

Axi

s

Ste

p 2

D

raw

th

e re

ctan

gle

• F

rom

th

e 8:

Sh

apes

men

u c

hoo

se 3

: Rec

tan

gle.

• C

lick

on

ce t

o de

fin

e th

e co

rner

of

the

rect

angl

e. T

hen

mov

e an

d cl

ick

agai

n. T

he

side

of

the

rect

angl

e is

now

def

ined

. Mov

e pe

rpen

dicu

larl

y to

dra

w t

he

rect

angl

e.

Cli

ck t

o an

chor

th

e sh

ape.

Ste

p 3

M

easu

re t

he

len

gth

s of

th

e si

des

of t

he

rect

angl

e.

• F

rom

th

e 7:

Mea

sure

men

t m

enu

ch

oose

1: L

engt

h (

Not

e th

at w

hen

you

scr

oll

over

th

e re

ctan

gle,

th

e va

lue

now

sh

own

is

the

peri

met

er o

f th

e re

ctan

gle.

)•

Sel

ect

each

en

dpoi

nt

of a

seg

men

t of

th

e re

ctan

gle.

Th

en c

lick

or

pres

s E

nte

r to

an

chor

th

e le

ngt

h o

f th

e se

gmen

t in

th

e w

ork

area

.•

Rep

eat

for

the

oth

er s

ides

of

the

rect

angl

e.

Exer

cise

s 1

. Wh

at a

ppea

rs t

o be

tru

e ab

out

the

oppo

site

sid

es o

f th

e re

ctan

gle?

2. D

raw

th

e di

agon

als

of t

he

rect

angl

e u

sin

g 5:

Seg

men

t fr

om t

he

6: P

oin

ts a

nd

Lin

es

Men

u. C

lick

on

tw

o op

posi

te v

erti

ces

to d

raw

th

e di

agon

al. R

epea

t to

dra

w t

he

oth

er

diag

onal

.a.

Mea

sure

eac

h d

iago

nal

usi

ng

the

mea

sure

men

t to

ol. W

hat

do

you

obs

erve

?b

. Wh

at i

s tr

ue

abou

t th

e tr

ian

gles

for

med

by

the

side

s of

th

e re

ctan

gle

and

a di

agon

al?

Just

ify

you

r co

ncl

usi

on.

3. P

ress

Cle

ar t

hre

e ti

mes

an

d se

lect

Yes

to

clea

r th

e sc

reen

. Rep

eat

the

step

s an

d dr

aw

anot

her

rec

tan

gle.

Do

the

rela

tion

ship

s th

at y

ou f

oun

d fo

r th

e fi

rst

rect

angl

e yo

u d

rew

h

old

tru

e fo

r th

is r

ecta

ngl

e?

6-4

T

he o

pp

osit

e s

ides o

f a r

ecta

ng

le a

re c

on

gru

en

t.

b

. T

he t

rian

gle

s a

re c

on

gru

en

t b

y S

SS

.

Y

es,

the o

pp

osit

e s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

co

ng

ruen

t.

a

. T

he d

iag

on

als

of

a r

ecta

ng

le a

re c

on

gru

en

t.

001_

042_

GE

OC

RM

C06

_890

515.

indd

295/

20/0

84:

51:0

0P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

30

G

lenc

oe G

eom

etry

Geo

met

er’s

Ske

tchp

ad A

ctiv

ity

Exp

lori

ng

Recta

ng

les

A q

uad

rila

tera

l w

ith

fou

r ri

ght

angl

es i

s a

rect

angl

e. T

he

Geo

met

er’s

Ske

tch

pad

is a

use

ful

tool

for

exp

lori

ng

som

e of

th

e ch

arac

teri

stic

s of

a r

ecta

ngl

e. U

se t

he

foll

owin

g st

eps

to d

raw

a

rect

angl

e.

Ste

p 1

U

se t

he

Lin

e to

ol t

o dr

aw a

lin

e an

ywh

ere

on t

he

scre

en.

Ste

p 2

U

se t

he

Poi

nt

tool

to

draw

a p

oin

t th

at i

s n

ot o

n t

he

lin

e. T

o dr

aw a

lin

e pe

rpen

dicu

lar

to t

he

firs

t li

ne

you

dre

w, s

elec

t th

e fi

rst

lin

e an

d th

e po

int.

Th

en c

hoo

se P

erp

end

icu

lar

Lin

e fr

om t

he

Con

stru

ct m

enu

.

Ste

p 3

U

se t

he

Poi

nt

tool

to

draw

a p

oin

t th

at i

s n

ot o

n e

ith

er o

f th

e li

nes

you

hav

e dr

awn

. R

epea

t th

e pr

oced

ure

in

Ste

p 2

to d

raw

li

nes

per

pen

dicu

lar

to t

he

two

lin

es y

ou

hav

e dr

awn

.

A r

ecta

ngl

e is

for

med

by

the

segm

ents

wh

ose

endp

oin

ts a

re t

he

poin

ts o

f in

ters

ecti

on o

f th

e li

nes

.

Exer

cise

sU

se t

he

mea

suri

ng

cap

abil

itie

s of

Th

e G

eom

eter

’s S

ket

chp

ad t

o ex

plo

re

the

char

acte

rist

ics

of a

rec

tan

gle.

1. W

hat

app

ears

to

be t

rue

abou

t th

e op

posi

te s

ides

of

the

rect

angl

e th

at y

ou d

rew

? M

ake

a co

nje

ctu

re a

nd

then

mea

sure

eac

h s

ide

to c

hec

k yo

ur

con

ject

ure

.

2. D

raw

th

e di

agon

als

of t

he

rect

angl

e by

usi

ng

the

Sel

ecti

on A

rrow

too

l to

ch

oose

tw

o op

posi

te v

erti

ces.

Th

en c

hoo

se S

egm

ent

from

th

e C

onst

ruct

men

u t

o dr

aw t

he

diag

onal

. Rep

eat

to d

raw

th

e ot

her

dia

gon

al.

a. M

easu

re e

ach

dia

gon

al. W

hat

do

you

obs

erve

?

b.

Wh

at i

s tr

ue

abou

t th

e tr

ian

gles

for

med

by

the

side

s of

th

e re

ctan

gle

and

a di

agon

al?

Just

ify

you

r co

ncl

usi

on.

3. C

hoo

se N

ew S

ket

ch f

rom

th

e F

ile

men

u a

nd

foll

ow s

teps

1–3

to

draw

an

oth

er

rect

angl

e. D

o th

e re

lati

onsh

ips

you

fou

nd

for

the

firs

t re

ctan

gle

you

dre

w h

old

tru

e fo

r th

is r

ecta

ngl

e al

so?

6 -4

T

he o

pp

osit

e s

ides o

f a r

ecta

ng

le a

re c

on

gru

en

t.

Th

e d

iag

on

als

of

a r

ecta

ng

le a

re c

on

gru

en

t.

Th

e t

rian

gle

s a

re c

on

gru

en

t b

y S

SS

.

Y

es,

the o

pp

osit

e s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

co

ng

ruen

t.

001_

042_

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OC

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_890

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17/0

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08:1

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M


NA

ME

DA

TE

PE

RIO

D

Lesson 6-5

Ch

ap

ter

6

31

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

Rh

om

bi

an

d S

qu

are

s

Pro

per

ties

of

Rh

om

bi a

nd

Sq

uar

es A

rh

omb

us

is a

qu

adri

late

ral

wit

h f

our

con

gru

ent

side

s. O

ppos

ite

side

s ar

e co

ngr

uen

t, s

o a

rhom

bus

is

also

a p

aral

lelo

gram

an

d h

as a

ll o

f th

e pr

oper

ties

of

a pa

rall

elog

ram

. R

hom

bi a

lso

hav

e th

e fo

llow

ing

prop

erti

es.

A s

qu

are

is a

par

alle

logr

am w

ith

fou

r co

ngr

uen

t si

des

and

fou

r co

ngr

uen

t an

gles

. A s

quar

e is

bot

h a

rec

tan

gle

and

a rh

ombu

s;

ther

efor

e, a

ll p

rope

rtie

s of

par

alle

logr

ams,

rec

tan

gles

, an

d rh

ombi

ap

ply

to s

quar

es.

In

rh

omb

us

AB

CD

, m∠

BA

C =

32.

Fin

d t

he

mea

sure

of

each

nu

mb

ered

an

gle.

AB

CD

is

a rh

ombu

s, s

o th

e di

agon

als

are

perp

endi

cula

r an

d �

AB

E

is a

rig

ht

tria

ngl

e. T

hu

s m

∠4

= 9

0 an

d m

∠1

= 9

0 -

32

or 5

8. T

he

diag

onal

s in

a r

hom

bus

bise

ct t

he

vert

ex a

ngl

es, s

o m

∠1

= m

∠2.

T

hu

s, m

∠2

= 5

8.A

rh

ombu

s is

a p

aral

lelo

gram

, so

the

oppo

site

sid

es a

re p

aral

lel.

∠B

AC

an

d ∠

3 ar

e al

tern

ate

inte

rior

an

gles

for

par

alle

l li

nes

, so

m∠

3 =

32.

Exer

cise

sQ

uad

rila

tera

l A

BC

D i

s a

rhom

bu

s. F

ind

eac

h v

alu

e or

mea

sure

.

1. I

f m

∠A

BD

= 6

0, f

ind

m∠

BD

C.

2. I

f A

E =

8, f

ind

AC

.

3. I

f A

B =

26

and

BD

= 2

0, f

ind

AE

. 4.

Fin

d m

∠C

EB

.

5. I

f m

∠C

BD

= 5

8, f

ind

m∠

AC

B.

6. I

f A

E =

3x

- 1

an

d A

C =

16,

fin

d x.

7. I

f m

∠C

DB

= 6

y an

d m

∠A

CB

= 2

y +

10,

fin

d y.

8. I

f A

D =

2x

+ 4

an

d C

D =

4x

- 4

, fin

d x.

MO

HR

B

CA

DEB

32°

12

34

C

AD

EB

6 -5

Th

e d

iag

on

als

are

pe

rpe

nd

icu

lar.

−−

−

M

H ⊥

−

−−

R

O

Ea

ch

dia

go

na

l b

ise

cts

a p

air o

f o

pp

osite

an

gle

s.

−−

−

M

H b

ise

cts

∠R

MO

an

d ∠

RH

O.

−−

−

R

O b

ise

cts

∠M

RH

an

d ∠

MO

H.

Exam

ple

60

16

24

90

32

3

10

4

001_

042_

GE

OC

RM

C06

_890

515.

indd

314/

11/0

811

:40:

54P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

32

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Rh

om

bi

an

d S

qu

are

s

Co

nd

itio

ns

for

Rh

om

bi a

nd

Sq

uar

es T

he

theo

rem

s be

low

can

hel

p yo

u p

rove

th

at

a pa

rall

elog

ram

is

a re

ctan

gle,

rh

ombu

s, o

r sq

uar

e.

If t

he

diag

onal

s of

a p

aral

lelo

gram

are

per

pen

dicu

lar,

th

en t

he

para

llel

ogra

m i

s a

rhom

bus.

If o

ne

diag

onal

of

a pa

rall

elog

ram

bis

ects

a p

air

of o

ppos

ite

angl

es, t

hen

th

e pa

rall

elog

ram

is

a r

hom

bus.

If o

ne

pair

of

con

secu

tive

sid

es o

f a

para

llel

ogra

m a

re c

ongr

uen

t, t

he

para

llel

ogra

m i

s a

rhom

bus.

If a

qu

adri

late

ral

is b

oth

a r

ecta

ngl

e an

d a

rhom

bus,

th

en i

t is

a s

quar

e.

D

eter

min

e w

het

her

par

alle

logr

am A

BC

D w

ith

ve

rtic

es A

(−3,

−3)

, B(1

, 1),

C(5

, −3)

, D(1

, −7)

is

a rh

ombu

s,

a re

cta

ngl

e, o

r a

squ

are

.

AC

= √

��

��

��

��

��

(-3

- 5

)2 +

((-

3 -

(-

3))2

= √

��

64 =

8

BD

= √

��

��

��

��

(1

-1)

2 +

(-

7 -

1)2

= √

��

64 =

8T

he

diag

onal

s ar

e th

e sa

me

len

gth

; th

e fi

gure

is

a re

ctan

gle.

Slo

pe o

f −

−

A

C =

-

3 -

(-

3)

−

-3

- 5

=

0 −

-8 =

8

Th

e li

ne

is h

oriz

onta

l.

Slo

pe o

f −

−−

B

D =

1

- (

-7)

−

1 -

1

=

8 −

0 = u

nd

efin

ed

Th

e li

ne

is v

erti

cal.

Sin

ce a

hor

izon

tal

and

vert

ical

lin

e ar

e pe

rpen

dicu

lar,

th

e di

agon

als

are

perp

endi

cula

r.

Par

alle

logr

am A

BC

D i

s a

squ

are

wh

ich

is

also

a r

hom

bus

and

a re

ctan

gle.

Exer

cise

sG

iven

eac

h s

et o

f ve

rtic

es, d

eter

min

e w

het

her

�A

BC

D i

s a

rhom

bus,

rec

tan

gle,

or

squ

are

. Lis

t al

l th

at a

pp

ly. E

xpla

in.

1. A

(0, 2

), B

(2, 4

), C

(4, 2

), D

(2, 0

) 2.

A(-

2, 1

), B

(-1,

3),

C(3

, 1),

D(2

, -1)

3. A

(-2,

-1)

, B(0

, 2),

C(2

, -1)

, D(0

, -4)

4.

A(-

3, 0

), B

(-1,

3),

C(5

, -1)

, D(3

, -4)

5. P

RO

OF

Wri

te a

tw

o-co

lum

n p

roof

.G

iven

: Par

alle

logr

am R

ST

U.

−−

R

S �

−−

S

T

Pro

ve: R

ST

U i

s a

rhom

bus.

Sta

tem

en

ts

Re

as

on

s

1.

RS

TU

is

a p

ara

lle

log

ram

1

. G

ive

n

−

−

RS

� −

−

ST

2. −

−

RS

� −

−

UT , −

−

RU

� −

−

ST

2.

De

fin

itio

n o

f a

pa

rall

elo

gra

m

3. −

−

UT �

−−

RS

� −

−

ST �

−−

RU

3

. S

ub

sti

tuti

on

4.

RS

TU

is

a r

ho

mb

us

4

. D

efi

nit

ion

of

a r

ho

mb

us

6 -5

Exam

ple

y

x

R

ec

tan

gle

, rh

om

bu

s,

sq

ua

re;

the

fo

ur

sid

es

are

� a

nd

co

ns

ec

uti

ve

s

ide

s a

re ⊥

.

R

ec

tan

gle

; c

on

se

cu

tiv

e s

ide

s a

re ⊥

.

R

ho

mb

us

; th

e f

ou

r s

ide

s a

re �

a

nd

co

ns

ec

uti

ve

sid

es

are

no

t ⊥

.

R

ec

tan

gle

; c

on

se

cu

tiv

e s

ide

s a

re ⊥

.

001_

042_

GE

OC

RM

C06

_890

515.

indd

324/

14/0

82:

58:3

7P

M

Lesson 6-5


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

33

G

lenc

oe G

eom

etry

Skill

s Pr

acti

ceR

ho

mb

i an

d S

qu

are

s

ALG

EBR

A Q

uad

rila

tera

l D

KL

M i

s a

rhom

bu

s.

1. I

f D

K=

8, f

ind

KL

.

2. I

f m

∠D

ML

= 8

2 fi

nd

m∠

DK

M.

3. I

f m

∠K

AL

= 2

x– 8

, fin

d x.

4. I

f D

A=

4x

and

AL

= 5

x– 3

, fin

d D

L.

5. I

f D

A=

4x

and

AL

= 5

x– 3

, fin

d A

D.

6. I

f D

M=

5y

+ 2

an

d D

K=

3y

+ 6

, fin

d K

L.

7.PR

OO

FW

rite

a t

wo-

colu

mn

pro

of.

Giv

en:R

ST

U i

s a

para

llel

ogra

m.

−−

−

RX

�−

−

TX

�−

−

SX

�−

−−

UX

P

rove

:RS

TU

is

a re

ctan

gle.

Pro

of

Sta

tem

ents

Rea

son

s

1.R

STU

is a

para

llelo

gra

m.

1.

Giv

en

−−

RX

� −

−

TX

� −

−

SX

� −

−

UX

2.

RX

= T

X =

SX

= U

X2.

Defi

nit

ion

of

co

ng

ruen

t seg

men

ts

3.

RX

+ X

T =

RT

, S

X +

XU

= S

U3.

Seg

Ad

d P

ostu

late

4.

RX

+X

T=

SU

4.

Su

bsti

tuti

on

Pro

pert

y

5.

RT =

SU

5.

Tra

nsit

ive P

rop

ert

y

6.

−−

RT �

−−

SU

6.

Defi

nit

ion

of

� s

eg

men

t

7.

RS

TU

is a

recta

ng

le.

7.

If d

iag

on

als

�,

the f

igu

re i

s a

re

cta

ng

le.

CO

OR

DIN

ATE

GEO

MET

RY

Giv

en e

ach

set

of

vert

ices

, det

erm

ine

wh

eth

er �

QR

ST

is a

rh

ombu

s, a

rec

tan

gle,

or

a sq

ua

re. L

ist

all

that

ap

ply

. Exp

lain

.

8. Q

(3, 5

), R

(3, 1

), S

(-1,

1),

T(-

1, 5

)

9. Q

(-5,

12)

, R(5

, 12)

, S(-

1, 4

), T

(-11

, 4)

10. Q

(-6,

-1)

, R(4

, -6)

, S(2

, 5),

T(-

8, 1

0)

11. Q

(2, -

4), R

(-6,

-8)

, S(-

10, 2

), T

(-2,

6)

ML

AD

K

6 -5

8

R

ho

mb

us,

recta

ng

le,

sq

uare

; all s

ides a

re c

on

gru

en

t an

d t

he

dia

go

nals

are

perp

en

dic

ula

r an

d c

on

gru

en

t.

R

ho

mb

us;

all s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

perp

en

dic

ula

r,

bu

t n

ot

co

ng

ruen

t.

R

ho

mb

us;

all s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

perp

en

dic

ula

r,

bu

t n

ot

co

ng

ruen

t.

N

on

e;

op

po

sit

e s

ides a

re c

on

gru

en

t, b

ut

the d

iag

on

als

are

neit

her

co

ng

ruen

t n

or

perp

en

dic

ula

r.

41

49

24

12

12

001_

042_

GE

OC

RM

C06

_890

515.

indd

335/

20/0

84:

51:1

4P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

34

G

lenc

oe G

eom

etry

Prac

tice

Rh

om

bi

an

d S

qu

are

s

PR

YZ

is

a rh

omb

us.

If

RK

= 5

, RY

= 1

3 an

d m

∠Y

RZ

= 6

7, f

ind

eac

h m

easu

re.

1. K

Y

2. P

K

3. m

∠Y

KZ

4. m

∠P

ZR

MN

PQ

is

a rh

omb

us.

If

PQ

= 3

√ �

2

an

d A

P =

3, f

ind

eac

h m

easu

re.

5. A

Q

6. m

∠A

PQ

7. m

∠M

NP

8. P

M

CO

OR

DIN

ATE

GEO

MET

RY

Giv

en e

ach

set

of

vert

ices

, det

erm

ine

wh

eth

er �

BE

FG

is

a r

hom

bus,

a r

ecta

ngl

e, o

r a

squ

are

. Lis

t al

l th

at a

pp

ly. E

xpla

in.

9. B

(-9,

1),

E(2

, 3),

F(1

2, -

2), G

(1, -

4)

10. B

(1, 3

), E

(7, -

3), F

(1, -

9), G

(-5,

-3)

11. B

(-4,

-5)

, E(1

, -5)

, F(-

2, -

1), G

(-7,

-1)

12. T

ESSE

LLA

TIO

NS

Th

e fi

gure

is

an e

xam

ple

of a

tes

sell

atio

n. U

se a

ru

ler

or p

rotr

acto

r to

mea

sure

th

e sh

apes

an

d th

en n

ame

the

quad

rila

tera

ls

use

d to

for

m t

he

figu

re.

K

P

YR

Z

AN M

QP

6 -5

R

ho

mb

us;

all s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

perp

en

dic

ula

r,

bu

t n

ot

co

ng

ruen

t.

R

ho

mb

us,

recta

ng

le,

sq

uare

; all s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

perp

en

dic

ula

r an

d c

on

gru

en

t.

R

ho

mb

us;

all s

ides a

re c

on

gru

en

t an

d t

he d

iag

on

als

are

perp

en

dic

ula

r,

bu

t n

ot

co

ng

ruen

t.

T

he f

igu

re c

on

sis

ts o

f 6 c

on

gru

en

t rh

om

bi.

12

12

90

67

45 90

63

001_

042_

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:41:

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M


NA

ME

DA

TE

PE

RIO

D

Lesson 6-5

Ch

ap

ter

6

35

G

lenc

oe G

eom

etry

1. T

RA

Y R

AC

KS

A t

ray

rack

loo

ks l

ike

a pa

rall

elog

ram

fro

m t

he

side

. Th

e le

vels

fo

r th

e tr

ays

are

even

ly s

pace

d.

Wh

at t

wo

labe

led

poin

ts f

orm

a r

hom

bus

wit

h b

ase

AA

' ?

2. S

LIC

ING

Ch

arle

s cu

ts a

rh

ombu

s al

ong

both

dia

gon

als.

He

ends

up

wit

h f

our

con

gru

ent

tria

ngl

es. C

lass

ify

thes

e tr

ian

gles

as

acu

te, o

btu

se, o

r ri

ght.

3. W

IND

OW

S T

he

edge

s of

a w

indo

w a

re

draw

n i

n t

he

coor

din

ate

plan

e.

Det

erm

ine

wh

eth

er t

he

win

dow

is

a sq

uar

e or

a r

hom

bus.

4.SQ

UA

RES

Mac

ken

zie

cut

a sq

uar

e al

ong

its

diag

onal

s to

get

fou

r co

ngr

uen

t ri

ght

tria

ngl

es. S

he

then

join

ed t

wo

of

them

alo

ng

thei

r lo

ng

side

s. S

how

th

at

the

resu

ltin

g sh

ape

is a

squ

are.

5. D

ESIG

N T

atia

nn

a m

ade

the

desi

gn

show

n. S

he

use

d 32

con

gru

ent

rhom

bi t

o cr

eate

th

e fl

ower

-lik

e de

sign

at

each

co

rner

.

a. W

hat

are

th

e an

gles

of

the

corn

er

rhom

bi?

b.

Wh

at k

inds

of

quad

rila

tera

ls a

re t

he

dott

ed a

nd

chec

kere

d fi

gure

s?

Wor

d Pr

oble

m P

ract

ice

Rh

om

bi

an

d S

qu

are

s

y

xO

AB

CDEFG

H

B,C,D,E,F,G

,H,

A20

inch

es

AB =

4 in

ches

,

6 -5

F a

nd

F′

rig

ht

tria

ng

les

rho

mb

us

Sam

ple

an

sw

er:

Th

e d

iag

on

als

of

a s

qu

are

cre

ate

fo

ur

co

ng

ruen

t ri

gh

t is

oscele

s t

rian

gle

s.

Wh

en

two

co

ng

ruen

t ri

gh

t is

oscele

s

tria

ng

les a

re j

oin

ed

alo

ng

th

eir

hyp

ote

nu

ses,

the r

esu

lt i

s a

q

uad

rila

tera

l w

ith

4 e

qu

al

sid

es

an

d 4

rig

ht

an

gle

co

rners

(b

ecau

se 4

5 +

45 =

90),

makin

g

it a

sq

uare

. 45,

135,

45,

135

Th

ey a

re a

ll s

qu

are

s.

001_

042_

GE

OC

RM

C06

_890

515.

indd

354/

11/0

811

:41:

19P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

36

G

lenc

oe G

eom

etry

Enri

chm

ent

Cre

ati

ng

Pyth

ag

ore

an

Pu

zzle

s

By

draw

ing

two

squ

ares

an

d cu

ttin

g th

em i

n a

cer

tain

way

, you

ca

n m

ake

a pu

zzle

th

at d

emon

stra

tes

the

Pyt

hag

orea

n T

heo

rem

. A

sam

ple

puzz

le i

s sh

own

. You

can

cre

ate

you

r ow

n p

uzz

le b

y fo

llow

ing

the

inst

ruct

ion

s be

low

.

1. C

aref

ull

y co

nst

ruct

a s

quar

e an

d la

bel

the

len

gth

of

a s

ide

as a

. Th

en c

onst

ruct

a s

mal

ler

squ

are

to

the

righ

t of

it

and

labe

l th

e le

ngt

h o

f a

side

as

b,

as s

how

n i

n t

he

figu

re a

bove

. Th

e ba

ses

shou

ld

be a

djac

ent

and

coll

inea

r.

2. M

ark

a po

int

X t

hat

is

b u

nit

s fr

om t

he

left

edg

e of

th

e la

rger

squ

are.

Th

en d

raw

th

e se

gmen

ts

from

th

e u

pper

lef

t co

rner

of

the

larg

er s

quar

e to

po

int

X, a

nd

from

poi

nt

X t

o th

e u

pper

rig

ht

corn

er

of t

he

smal

ler

squ

are.

3. C

ut

out

and

rear

ran

ge y

our

five

pie

ces

to f

orm

a

larg

er s

quar

e. D

raw

a d

iagr

am t

o sh

ow y

our

answ

er.

4. V

erif

y th

at t

he

len

gth

of

each

sid

e is

equ

al t

o

√ �

��

a2 +

b2 .

a

a

bX

b

b

6 -5

See s

tud

en

ts’

wo

rk.

A s

am

ple

an

sw

er

is s

ho

wn

.

001_

042_

GE

OC

RM

C06

_890

515.

indd

364/

11/0

811

:41:

24P

M

Lesson 6-6


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

37

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

Tra

pezo

ids a

nd

Kit

es

Pro

per

ties

of

Trap

ezo

ids

A t

rap

ezoi

d i

s a

quad

rila

tera

l w

ith

ex

actl

y on

e pa

ir o

f pa

rall

el s

ides

. Th

e m

idse

gmen

t or

med

ian

of

a t

rape

zoid

is

the

segm

ent

that

con

nec

ts t

he

mid

poin

ts o

f th

e le

gs o

f th

e tr

apez

oid.

Its

mea

sure

is

equ

al t

o on

e-h

alf

the

sum

of

the

len

gth

s of

th

e ba

ses.

If

the

legs

are

con

gru

ent,

th

e tr

apez

oid

is

an i

sosc

eles

tra

pez

oid

. In

an

iso

scel

es t

rape

zoid

bot

h p

airs

of

bas

e an

gles

are

con

gru

ent

and

the

diag

onal

s ar

e co

ngr

uen

t.

T

he

vert

ices

of

AB

CD

are

A(-

3, -

1), B

(-1,

3),

C

(2, 3

), a

nd

D(4

, -1)

. Sh

ow t

hat

AB

CD

is

a tr

apez

oid

an

d

det

erm

ine

wh

eth

er i

t is

an

iso

scel

es

trap

ezoi

d.

slop

e of

−−

A

B =

3

- (

-1)

−

-1

- (

-3)

= 4 −

2 = 2

slop

e of

−−

−

A

D =

-1

- (

-1)

−

4 -

(-

3)

= 0 −

7 = 0

slop

e of

−−

−

B

C =

3

- 3

−

2 -

(-

1) =

0 −

3 = 0

slop

e of

−−

−

C

D =

-1

- 3

−

4 -

2

= -

4 −

2 =

-2

Exa

ctly

tw

o si

des

are

para

llel

, −−

−

A

D a

nd

−−

−

B

C , s

o A

BC

D i

s a

trap

ezoi

d. A

B =

CD

, so

AB

CD

is

an i

sosc

eles

tra

pezo

id.

Exer

cise

sF

ind

eac

h m

easu

re.

1. m

∠D

55

2.

m∠

L

140

125°

5

5

40°

CO

OR

DIN

ATE

GEO

MET

RY

For

eac

h q

uad

rila

tera

l w

ith

th

e gi

ven

ver

tice

s, v

erif

y th

at t

he

qu

adri

late

ral

is a

tra

pez

oid

an

d d

eter

min

e w

het

her

th

e fi

gure

is

an

isos

cele

s tr

apez

oid

.

3. A

(−1,

1),

B(3

, 2),

C(1

,−2)

, D(−

2, −

1)

4. J

(1, 3

), K

(3, 1

), L

(3, −

2), M

(−2,

3)

−

−

AD

‖ −

−

BC

, −

−

AB

∦ −

−

CD

, A

BC

D i

s a

−−

JK

‖ −

−

LM

, −

−

JM

∦ −

−

KL

, JK

LM

is a

ntr

ap

ezo

id b

ut

no

t an

iso

scele

s

iso

scele

s t

rap

ezo

id b

ecau

se

trap

ezo

id b

ecau

se A

B =

√ �

�

17 ,

J

M =

KL =

3.

CD

= √

��

10 .

For

tra

pez

oid

HJ

KL

, M a

nd

N a

re t

he

mid

poi

nts

of

the

legs

.

5.

If

HJ

= 3

2 an

d L

K =

60,

fin

d M

N.

46

6.

If

HJ

= 1

8 an

d M

N =

28,

fin

d L

K.

38

T

RU

base

base

Sle

g

x

y

O

B

DA

C

6-6

STU

R is a

n isoscele

s t

rapezoid

.

−−

S

R �

−−

TU

; ∠

R �

∠U

, ∠

S �

∠T

Exam

ple

AB

=

√ �

��

��

��

��

�

(-3

- (

-1)

)2 +

(-

1 -

3)2

=

√

��

�

4 +

16

= √

��

20 =

2 √

�

5

CD

= √

��

��

��

��

�

(2

- 4

)2 +

(3

- (

-1)

)2

=

√ �

��

4 +

16

= √

��

20 =

2 √

�

5

LK

NM

HJ

001_

042_

GE

OC

RM

C06

_890

515.

indd

374/

11/0

811

:41:

28P

M



Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

38

G

lenc

oe G

eom

etry

Pro

per

ties

of

Kit

es A

kit

e is

a q

uad

rila

tera

l w

ith

exa

ctly

tw

o pa

irs

of c

onse

cuti

ve

con

gru

ent

side

s. U

nli

ke a

par

alle

logr

am, t

he

oppo

site

sid

es o

f a

kite

are

not

con

gru

ent

or p

aral

lel.

Th

e di

agon

als

of a

kit

e ar

e pe

rpen

dicu

lar.

For

kit

e R

MN

P,

−−

−

M

P ⊥

−−

−

R

N

In a

kit

e, e

xact

ly o

ne

pair

of

oppo

site

an

gles

is

con

gru

ent.

For

kit

e R

MN

P, ∠

M �

∠P

If

WX

YZ

is

a k

ite,

fin

d m

∠Z

.

Th

e m

easu

res

of ∠

Y a

nd

∠W

are

not

con

gru

ent,

so

∠X

� ∠

Z.

m∠

X +

m

∠Y

+ m

∠Z

+ m

∠W

= 36

0 m

∠X

+ 60

+ m

∠Z

+ 80

= 36

0m

∠X

+ m

∠Z

= 22

0m

∠X

= 11

0, m

∠Z

= 11

0

If

AB

CD

is

a k

ite,

fin

d B

C.

Th

e di

agon

als

of a

kit

e ar

e pe

rpen

dicu

lar.

Use

th

e P

yth

agor

ean

Th

eore

m t

o fi

nd

the

mis

sin

g le

ngt

h.

BP

2 +

PC

2 =

BC

2

52 +

122

= B

C2

169

= B

C2

13 =

BC

Exer

cise

sIf

GH

JK

is

a k

ite,

fin

d e

ach

mea

sure

.

1. F

ind

m∠

JR

K.

90

2. I

f R

J =

3 a

nd

RK

= 1

0, f

ind

JK

. √

��

109

3. I

f m

∠G

HJ

= 9

0 a

nd

m∠

GK

J =

1

10, f

ind

m∠

HG

K.

80

4. I

f H

J =

7, f

ind

HG

. 7

5. I

f H

G =

7 a

nd

GR

= 5

, fin

d H

R.

√ �

�

24 =

2 √

�

6

6. I

f m

∠G

HJ

= 5

2 an

d m

∠G

KJ

= 9

5, f

ind

m∠

HG

K.

106.5

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Tra

pezo

ids a

nd

Kit

es

6-6

Exam

ple

1

Exam

ple

2

80°

60°

5 5

4P

12

001_

042_

GE

OC

RM

C06

_890

515.

indd

384/

11/0

811

:41:

36P

M

Lesson 6-6


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

39

G

lenc

oe G

eom

etry

ALG

EBR

A F

ind

eac

h m

easu

re.

1. m

∠S

117

2.

m∠

M

38

63°

1414

142°

2121

3. m

∠D

127

4.

RH

√

��

544 =

4 √

��

34

36°

70°

20

12 12

ALG

EBR

A F

or t

rap

ezoi

d H

JK

L, T

an

d S

are

mid

poi

nts

of

the

legs

.

5. I

f H

J =

14

and

LK

= 4

2, f

ind

TS

. 28

6. I

f L

K =

19

and

TS

= 1

5, f

ind

HJ

. 11

7. I

f H

J =

7 a

nd

TS

= 1

0, f

ind

LK

. 13

8. I

f K

L =

17

and

JH

= 9

, fin

d S

T.

13

CO

OR

DIN

ATE

GEO

MET

RY

EF

GH

is

a q

uad

rila

tera

l w

ith

ver

tice

s E

(1, 3

), F

(5, 0

),

G(8

, −5)

, H(−

4, 4

).

9. V

erif

y th

at E

FG

H i

s a

trap

ezoi

d.

−−

EF

‖ −

−

GH

, b

ut

−−

HE

∦ −

−

FG

10. D

eter

min

e w

het

her

EF

GH

is

an i

sosc

eles

tra

pezo

id. E

xpla

in.

n

ot

iso

scele

s;

EH

= √

��

26 a

nd

FG

= √

��

34

6-6

Skill

s Pr

acti

ceTra

pezo

ids a

nd

Kit

es

001_

042_

GE

OC

RM

C06

_890

515.

indd

395/

17/0

86:

09:0

8P

M



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

PDF Pass



NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

40

G

lenc

oe G

eom

etry

Prac

tice

Tra

pezo

ids a

nd

Kit

es

Fin

d e

ach

mea

sure

.

1. m

∠T

60

2.

m∠

Y

112

60°

68°

7 7

3. m

∠Q

101

4.

BC

√

��

65

110°

48°

114

7 7

ALG

EBR

A F

or t

rap

ezoi

d F

ED

C, V

an

d Y

are

mid

poi

nts

of

the

legs

.

5. I

f F

E =

18

and

VY

= 2

8, f

ind

CD

. 38

6. I

f m

∠F

= 1

40 a

nd

m∠

E =

125

, fin

d m

∠D

. 55

CO

OR

DIN

ATE

GEO

MET

RY

RS

TU

is

a q

uad

rila

tera

l w

ith

ver

tice

s R

(−3,

−3)

, S(5

, 1),

T

(10,

−2)

, U(−

4, −

9).

7. V

erif

y th

at R

ST

U i

s a

trap

ezoi

d.

−−

RS

‖ −

−

TU

8. D

eter

min

e w

het

her

RS

TU

is

an i

sosc

eles

tra

pezo

id. E

xpla

in.

n

ot

iso

scele

s;

RU

= √

��

37 a

nd

ST

= √

��

34

9. C

ON

STR

UC

TIO

N A

set

of

stai

rs l

eadi

ng

to t

he

entr

ance

of

a bu

ildi

ng

is d

esig

ned

in

th

e sh

ape

of a

n i

sosc

eles

tra

pezo

id w

ith

th

e lo

nge

r ba

se a

t th

e bo

ttom

of

the

stai

rs a

nd

the

shor

ter

base

at

the

top.

If

the

bott

om o

f th

e st

airs

is

21 f

eet

wid

e an

d th

e to

p is

14

feet

w

ide,

fin

d th

e w

idth

of

the

stai

rs h

alfw

ay t

o th

e to

p.

10. D

ESK

TO

PS A

car

pen

ter

nee

ds t

o re

plac

e se

vera

l tr

apez

oid-

shap

ed d

eskt

ops

in a

cl

assr

oom

. Th

e ca

rpen

ter

know

s th

e le

ngt

hs

of b

oth

bas

es o

f th

e de

skto

p. W

hat

oth

er

mea

sure

men

ts, i

f an

y, d

oes

the

carp

ente

r n

eed?

6-6

FE

DC

VY

17.5

ft

Sam

ple

an

sw

er:

th

e m

easu

res o

f th

e b

ase a

ng

les

001_

042_

GE

OC

RM

C06

_890

515.

indd

404/

14/0

83:

04:5

3P

M

Lesson 6-6


NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

41

G

lenc

oe G

eom

etry

Wor

d Pr

oble

m P

ract

ice

Tra

pezo

ids a

nd

Kit

es

1. P

ERSP

ECTI

VE

Art

ists

use

dif

fere

nt

tech

niq

ues

to

mak

e th

ings

app

ear

to b

e 3-

dim

ensi

onal

wh

en d

raw

ing

in t

wo

dim

ensi

ons.

Kev

in d

rew

th

e w

alls

of

a ro

om. I

n r

eal

life

, all

of

the

wal

ls a

re

rect

angl

es. I

n w

hat

sh

ape

did

he

draw

th

e si

de w

alls

to

mak

e th

em a

ppea

r 3-

dim

ensi

onal

?

2.PL

AZA

In

ord

er t

o gi

ve t

he

feel

ing

of

spac

iou

snes

s, a

n a

rch

itec

t de

cide

s to

m

ake

a pl

aza

in t

he

shap

e of

a k

ite.

T

hre

e of

th

e fo

ur

corn

ers

of t

he

plaz

a ar

e sh

own

on

th

e co

ordi

nat

e pl

ane.

If

the

fou

rth

cor

ner

is

in t

he

firs

t qu

adra

nt,

wh

at a

re i

ts c

oord

inat

es?

y

x

y

x

(6,

1)

3. A

IRPO

RTS

A s

impl

ifie

d dr

awin

g of

th

e re

ef r

un

way

com

plex

at

Hon

olu

lu

Inte

rnat

ion

al A

irpo

rt i

s sh

own

bel

ow.

How

man

y tr

apez

oids

are

th

ere

in t

his

im

age?

4. L

IGH

TIN

G A

lig

ht

outs

ide

a ro

om s

hin

es

thro

ugh

th

e do

or a

nd

illu

min

ates

a

trap

ezoi

dal

regi

on A

BC

D o

n t

he

floo

r.

AB

DC

Un

der

wh

at c

ircu

mst

ance

s w

ould

tr

apez

oid

AB

CD

be

isos

cele

s?

5. R

ISER

S A

ris

er i

s de

sign

ed t

o el

evat

e a

spea

ker.

Th

e ri

ser

con

sist

s of

4

trap

ezoi

dal

sect

ion

s th

at c

an b

e st

acke

d on

e on

top

of

the

oth

er t

o pr

odu

ce t

rape

zoid

s of

var

yin

g h

eigh

ts.

20 fe

et

10 fe

et

All

of

the

stag

es h

ave

the

sam

e h

eigh

t.

If a

ll f

our

stag

es a

re u

sed,

th

e w

idth

of

the

top

of t

he

rise

r is

10

feet

.

a. I

f on

ly t

he

bott

om t

wo

stag

es a

re

use

d, w

hat

is

the

wid

th o

f th

e to

p of

th

e re

sult

ing

rise

r?

b.

Wh

at w

ould

be

the

wid

th o

f th

e ri

ser

if t

he

bott

om t

hre

e st

ages

are

use

d?

6-6

trap

ezo

ids

W

hen

th

e l

igh

t so

urc

e i

s a

n e

qu

al

dis

tan

ce f

rom

C a

nd

D,

sh

inin

g

str

aig

ht

thro

ug

h t

he d

oo

r.

15 f

t

12.5

ft5

001_

042_

GE

OC

RM

C06

_890

515.

indd

414/

14/0

83:

05:0

5P

M



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ivision o

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-Hill C

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NA

ME

DA

TE

PE

RIO

D

Ch

ap

ter

6

42

G

lenc

oe G

eom

etry

Enri

chm

ent

Qu

ad

rila

tera

ls i

n C

on

str

ucti

on

Qu

adri

late

rals

are

oft

en u

sed

in c

onst

ruct

ion

wor

k.

1. T

he

diag

ram

at

the

righ

t re

pres

ents

a r

oof

fr

ame

and

show

s m

any

quad

rila

tera

ls. F

ind

the

foll

owin

g sh

apes

in

th

e di

agra

m a

nd

shad

e in

th

eir

edge

s.

a. i

sosc

eles

tri

angl

e

b.

scal

ene

tria

ngl

e

c. r

ecta

ngl

e

d.

rhom

bus

e. t

rape

zoid

(n

ot i

sosc

eles

)

f. is

osce

les

trap

ezoi

d

2. T

he

figu

re a

t th

e ri

ght

repr

esen

ts a

win

dow

. Th

e

woo

den

par

t be

twee

n t

he

pan

es o

f gl

ass

is 3

in

ches

w

ide.

Th

e fr

ame

arou

nd

the

oute

r ed

ge i

s 9

inch

es

wid

e. T

he

outs

ide

mea

sure

men

ts o

f th

e fr

ame

are

60 i

nch

es b

y 81

in

ches

. Th

e h

eigh

t of

th

e to

p an

d bo

ttom

pan

es i

s th

e sa

me.

Th

e to

p th

ree

pan

es a

re

the

sam

e si

ze.

a. H

ow w

ide

is t

he

bott

om p

ane

of g

lass

?

b.

How

wid

e is

eac

h t

op p

ane

of g

lass

?

c. H

ow h

igh

is

each

pan

e of

gla

ss?

3. E

ach

edg

e of

th

is b

ox h

as b

een

rei

nfo

rced

w

ith

a p

iece

of

tape

. Th

e bo

x is

10

inch

es

hig

h, 2

0 in

ches

wid

e, a

nd

12 i

nch

es d

eep.

W

hat

is

the

len

gth

of

the

tape

th

at h

as

been

use

d?

jack

rafte

rs

hip

rafte

r

com

mon

rafte

rs

Ro

of

Fra

me

ridge

boa

rdva

lley

rafte

r

plat

e

60 in

.

81 in

.

9 in

.

3 in

.

3 in

.

9 in

.

9 in

.

10 in

.

12 in

.

20 in

.

6-6

See s

tud

en

ts’

wo

rk.

42 i

n.

12 i

n.

30 i

n.

168 i

n.

001_

042_

GE

OC

RM

C06

_890

515.

indd

424/

11/0

811

:42:

09P

M



Copyright

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An

swer

s


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Chapter 6 Assessment Answer Key

Quiz 2 (Lesson 6-3)

Page 45

Quiz 1 (Lessons 6-1 and 6-2) Quiz 3 (Lessons 6-4 and 6-5) Mid-Chapter TestPage 45 Page 46 Page 47

Quiz 4 (Lesson 6-6)

Page 46

1.

2.

3.

4.

5.

12,240

45

20

(-1, 10)

A

1.

2.

3.

4.

5.

No; none of the

tests for �s are fulfi lled.

true

false

true

28 cm

1.

2.

3.

4.

5.

B

65

115

true

rectangle, rhombus, square

1.

2.

3.

4.

5.

118

√ �� 50 = 5 √ � 2

5

15.5

Use the distance formula to show CF = DE.

1.

2.

3.

4.

5.

B

J

A

G

C

6.

7.

8.

9.

10.

50

42

yes; opp. sides are � to each other

30, 150

No; slope −−

XY = - 3 −

5

and slope of −−

WZ = - 1 −

3 ,

so opposite sides are

not parallel.


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Chapter 6 Assessment Answer KeyVocabulary Test Form 1

Page 48 Page 49 Page 50

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

isosceles trapezoid

parallelogram

trapezoid

square

rhombus

false, rectangle

false; rhombus

diagonals

median

Sample answer:

angles formed by the

base and one of the

legs of the trapezoid

the nonparallel sides of a trapezoid

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

H

B

H

D

F

B

F

D

F

C

12.

13.

14.

15.

16.

17.

18.

19.

20.

H

B

J

A

G

A

J

A

G

B:

x = 7, m∠WYZ = 41


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Chapter 6 Assessment Answer KeyForm 2A Form 2B

Page 51 Page 52 Page 53 Page 54

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

G

D

F

C

F

A

H

B

F

D

12.

13.

14.

15.

16.

17.

18.

19.

20.

J

C

H

A

G

D

H

D

G

B: 7 or -4

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

B

J

D

F

B

G

A

H

C

J

B

12.

13.

14.

15.

16.

17.

18.

19.

20.

G

D

G

D

H

C

F

A

H

B: 22


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Chapter 6 Assessment Answer KeyForm 2C

Page 55 Page 56

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

1080

19

40

18

8

122

(6, 4)

Yes; −−

AB and −−

CD are

|| and �.

No; the slopes are

9 −

4 ,

1 −

7 , 1, and

2 −

3 .

Thus, ABCD doesnot have || sides.

- 2 −

3

22

Yes; if the diagonals

of a � are �, then

the � is a rectangle.

67.5

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

(4, 0)

31

Yes; ABCD has only one pair of opposite sides ||,

−−

BC and −−

AD .

6

26

true

true

true

false

true

true

false

x = 9, y = 2


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Chapter 6 Assessment Answer KeyForm 2D

Page 57 Page 58

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

360

38

90

50

3.6

117

( 1 − 2 , 3)

Yes; Both pairs of

opp. sides are �.

ABCD has two pairs of

|| sides, −−

AB || −−

CD

and −−

BD || −−

DA ; it is a �.

-4

8

One rt. ∠ means that the other � will be rt. �. If all 4 � are

rt. �, the � is a rectangle.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

B:

72

(-3, 1)

16

Yes; ABCD has only

one pair of opp.

sides ||, −−

AB and −−

BD .

8

17

false

true

false

false

true

false

true

90


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Chapter 6 Assessment Answer KeyForm 3

Page 59 Page 60

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

7

30; 30, 47, 120,

179, 174, and 170

180 −

x

65

8 or 32

Yes; the diagonals bisect each other.

slope of −−

CD = 2 −

3 ;

slope of −−

DA = -2.

28

72

8 √ � 2

4

9

13

14.

15.

16.

17.

18.

19.

20.

B:

No; two pairs of congruent consecutive

sides do not exist.

CD = √ �� 72 , DA = √ �� 65 ,

AB = √ �� 20 , BC = √ �� 17

Yes; −−

AB ⊥ −−

BC ,

−−

BC ⊥ −−

CD , −−

CD ⊥ −−

AD ,

so opp � are �, and

all � are rt. �.

ABCD has 2 pairs of

opp. sides �, −−

AB

� −−

CD and −−

BC � −−

DA ,

so ABCD is a �.

105

Opp. sides of a � are �.

If both pairs of opp. sides of a

quad. are �, then the quad.

is a �.

27

54


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Chapter 6 Assessment Answer Key Extended-Response Test, Page 61 Scoring Rubric

Score General Description Specifi c Criteria

4 Superior

A correct solution that

is supported by well-

developed, accurate

explanations

• Shows thorough understanding of the concepts of angles of polygons, properties of parallelograms, rectangles, rhombi, squares, and trapezoids.

• Uses appropriate strategies to solve problems.

• Computations are correct.

• Written explanations are exemplary.

• Graphs and fi gures are accurate and appropriate.

• Goes beyond requirements of some or all problems.

3 Satisfactory

A generally correct solution,

but may contain minor fl aws

in reasoning or computation

• Shows an understanding of the concepts of angles of polygons, properties of parallelograms, rectangles, rhombi, squares, and trapezoids.

• Uses appropriate strategies to solve problems.

• Computations are mostly correct.

• Written explanations are effective.

• Graphs and fi gures are mostly accurate and appropriate.

• Satisfi es all requirements of problems.

2 Nearly Satisfactory

A partially correct

interpretation and/or solution

to the problem

• Shows an understanding of most of the concepts of angles of polygons, properties of parallelograms, rectangles, rhombi, squares, and trapezoids.

• May not use appropriate strategies to solve problems.

• Computations are mostly correct.

• Written explanations are satisfactory.

• Graphs and fi gures are mostly accurate.

• Satisfi es the requirements of most of the problems.

1 Nearly Unsatisfactory

A correct solution with no

supporting evidence or

explanation

• Final computation is correct.

• No written explanations or work shown to substantiate the fi nal

computation.

• Graphs and fi gures may be accurate but lack detail or explanation.

• Satisfi es minimal requirements of some of the problems.

0 Unsatisfactory

An incorrect solution

indicating no mathematical

understanding of the concept

or task, or no solution is

given

• Shows little or no understanding of most of the concepts of angles of polygons, properties of parallelograms, rectangles, rhombi, squares, and trapezoids.

• Does not use appropriate strategies to solve problems.

• Computations are incorrect.

• Written explanations are unsatisfactory.

• Graphs and fi gures are inaccurate or inappropriate.

• Does not satisfy requirements of problems.

• No answer may be given.


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Chapter 6 Assessment Answer Key Extended-Response Test, Page 61

Sample Answers

1. a. Any type of convex polygon can be drawn as long as one is regular and one is not regular and both have the same number of sides.

90°

70°

60°

110°120°

90°

90°

90°

b. Check to be sure that the exterior angles have been properly drawn and accurately measured.

c. 4(90) = 360; 120 + 70 + 60 + 110 = 360; The sum of the exterior angles of each figure should be 360°. The sum of the exterior angles of both the regular convex polygon and the irregular convex polygon is 360°.

2. The student should draw a rectangle and join the midpoints of consecutive sides. The figure formed inside is a rhombus. Since all four small triangles can be proved to be congruent by SAS, the four sides of the interior quadrilateral are congruent by CPCTC, making it a rhombus.

3. The student should draw an isosceles trapezoid with one pair of opposite sides parallel and the other pair of opposite sides congruent, as in the figure below.

4. a. Possible properties:A square has four congruent sides and a rectangle may not.A square has perpendicular diagonals and a rectangle may not.The diagonals of a square bisect the angles and those in a rectangle may not.

b. Possible properties:A square has four right angles and a rhombus may not.The diagonals of a square are congruent and those of a rhombus may not be.

c. Possible properties:A rectangle has four right angles and a parallelogram may not.The diagonals of a rectangle are congruent and those of a parallelogram may not be.

In addition to the scoring rubric found on page A30, the following sample answers

may be used as guidance in evaluating open-ended assessment items.


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Chapter 6 Assessment Answer KeyStandardized Test Practice

Page 62 Page 63

8. F G H J

9. A B C D

10. F G H J

11. A B C D

12. F G H J

13.

9

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7

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14.

9

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1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D


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PDF Pass

Chapter 6 Assessment Answer KeyStandardized Test Practice (continued)

Page 64

15.

16.

17.

18.

19.

20.

21. a.

b.

c.

hexagon;concave;irregular

88.9 cm

−−

DL

x = 11, y = 2

Assume thatneither bell costmore than $45.

40 cm3

−−

ML

−−

JL

J


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Chapter 6 Assessment Answer KeyUnit 2 Test

Page 65 Page 66

11.

12.

13.

14.

15.

16.

17.

18.

19.

yes

7 −

2 < x < 31

−

2

9

m∠JHK = 52;

m∠HMK = 108,

and x = 8

No; opp. side are

not ||.

5

5

11

In a parallelogram,

opposite sides are

congruent. Using

the distance

formula, PQ = √ �� 20 ,

PS = √ �� 20 , QR = √ � 8 ,RS = √ � 8 .

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

right; obtuse;

acute

m∠1 = 25; m∠2 = 25; m∠3 = 130

−−

AF � −−

ST , −−

FP � −−

TX , −−

PA � −−

XS

yes

ASA Postulate

∠A � ∠C

111

a = 9;

m∠ZWT = 32

∠YWZ

Neither appliance cost

more than $603.


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