chapter 8 resource masters - · pdf filechapter 8 test, form 2d . . . . . . . . . . ....

Chapter 8Resource Masters

Geometry

Consumable WorkbooksMany of the worksheets contained in the Chapter Resource Masters bookletsare available as consumable workbooks.

Study Guide and Intervention Workbook 0-07-860191-6Skills Practice Workbook 0-07-860192-4Practice Workbook 0-07-860193-2Reading to Learn Mathematics Workbook 0-07-861061-3

ANSWERS FOR WORKBOOKS The answers for Chapter 8 of these workbookscan be found in the back of this Chapter Resource Masters booklet.

Copyright © by The McGraw-Hill Companies, Inc. All rights reserved.Printed in the United States of America. 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’s Geometry. Any other reproduction, for use or sale, is prohibited without prior written permission of the publisher.

Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027

ISBN: 0-07-860185-1 GeometryChapter 8 Resource Masters

1 2 3 4 5 6 7 8 9 10 009 11 10 09 08 07 06 05 04 03

© Glencoe/McGraw-Hill iii Glencoe Geometry

Contents

Vocabulary Builder . . . . . . . . . . . . . . . . vii

Proof Builder . . . . . . . . . . . . . . . . . . . . . . ix

Lesson 8-1Study Guide and Intervention . . . . . . . . 417–418Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 419Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 420Reading to Learn Mathematics . . . . . . . . . . 421Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 422







Chapter 8 AssessmentChapter 8 Test, Form 1 . . . . . . . . . . . . 459–460Chapter 8 Test, Form 2A . . . . . . . . . . . 461–462Chapter 8 Test, Form 2B . . . . . . . . . . . 463–464Chapter 8 Test, Form 2C . . . . . . . . . . . 465–466Chapter 8 Test, Form 2D . . . . . . . . . . . 467–468Chapter 8 Test, Form 3 . . . . . . . . . . . . 469–470Chapter 8 Open-Ended Assessment . . . . . . 471Chapter 8 Vocabulary Test/Review . . . . . . . 472Chapter 8 Quizzes 1 & 2 . . . . . . . . . . . . . . . 473Chapter 8 Quizzes 3 & 4 . . . . . . . . . . . . . . . 474Chapter 8 Mid-Chapter Test . . . . . . . . . . . . 475Chapter 8 Cumulative Review . . . . . . . . . . . 476Chapter 8 Standardized Test Practice . 477–478

Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1

ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A32

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 8 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 8 Resource Masters includes the core materials neededfor Chapter 8. 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 in theGeometry TeacherWorks CD-ROM.

Vocabulary Builder Pages vii–viiiinclude a student study tool that presentsup to twenty of the key vocabulary termsfrom the chapter. Students are to recorddefinitions and/or examples for each term.You may suggest that students highlight orstar the terms with which they are notfamiliar.

WHEN TO USE Give these pages tostudents before beginning Lesson 8-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toadd definitions and examples as theycomplete each lesson.

Vocabulary Builder Pages ix–xinclude another student study tool thatpresents up to fourteen of the key theoremsand postulates from the chapter. Studentsare to write each theorem or postulate intheir own words, including illustrations ifthey choose to do so. You may suggest thatstudents highlight or star the theorems orpostulates with which they are not familiar.

WHEN TO USE Give these pages tostudents before beginning Lesson 8-1.Encourage them to add these pages to theirGeometry Study Notebook. Remind them toupdate it as they complete each lesson.

Study Guide and InterventionEach lesson in Geometry addresses twoobjectives. There is one Study Guide andIntervention master for each objective.

WHEN TO USE Use these masters asreteaching activities for students who needadditional reinforcement. These pages canalso be used in conjunction with the StudentEdition as an instructional tool for studentswho have been absent.

Skills Practice There is one master foreach lesson. These provide computationalpractice at a basic level.

WHEN TO USE These masters can be used with students who have weakermathematics backgrounds or needadditional reinforcement.

Practice There is one master for eachlesson. These problems more closely followthe structure of the Practice and Applysection of the Student Edition exercises.These exercises are of average difficulty.

WHEN TO USE These provide additionalpractice options or may be used ashomework for second day teaching of thelesson.

Reading to Learn MathematicsOne master is included for each lesson. Thefirst section of each master asks questionsabout the opening paragraph of the lessonin the Student Edition. Additionalquestions ask students to interpret thecontext of and relationships among termsin the lesson. Finally, students are asked tosummarize what they have learned usingvarious representation techniques.

WHEN TO USE This master can be usedas a study tool when presenting the lessonor as an informal reading assessment afterpresenting the lesson. It is also a helpfultool for ELL (English Language Learner)students.

© Glencoe/McGraw-Hill v Glencoe Geometry

Enrichment There is one extensionmaster for each lesson. These activities mayextend the concepts in the lesson, offer anhistorical or multicultural look at theconcepts, or widen students’ perspectives onthe mathematics they are learning. Theseare not written exclusively for honorsstudents, but are accessible for use with alllevels of students.

WHEN TO USE These may be used asextra credit, short-term projects, or asactivities for days when class periods areshortened.

Assessment OptionsThe assessment masters in the Chapter 8Resources Masters offer a wide range ofassessment tools for intermediate and finalassessment. The following lists describe eachassessment master and its intended use.

Chapter Assessment CHAPTER TESTS• Form 1 contains multiple-choice questions

and is intended for use with basic levelstudents.

• Forms 2A and 2B contain multiple-choicequestions aimed at the average levelstudent. These tests are similar in formatto offer comparable testing situations.

• Forms 2C and 2D are composed of free-response questions aimed at the averagelevel student. These tests are similar informat to offer comparable testingsituations. Grids with axes are providedfor questions assessing graphing skills.

• Form 3 is an advanced level test withfree-response questions. Grids withoutaxes are provided for questions assessinggraphing skills.

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

• The Open-Ended Assessment includesperformance assessment tasks that aresuitable for all students. A scoring rubricis included for evaluation guidelines.Sample answers are provided forassessment.

• A Vocabulary Test, suitable for allstudents, includes a list of the vocabularywords in the chapter and ten questionsassessing students’ knowledge of thoseterms. This can also be used in conjunc-tion with one of the chapter tests or as areview worksheet.

Intermediate Assessment• Four free-response quizzes are included

to offer assessment at appropriateintervals in the chapter.

• A Mid-Chapter Test provides an optionto assess the first half of the chapter. It iscomposed of both multiple-choice andfree-response questions.

Continuing Assessment• The Cumulative Review provides

students an opportunity to reinforce andretain skills as they proceed throughtheir study of Geometry. It can also beused as a test. This master includes free-response questions.

• The Standardized Test Practice offerscontinuing review of geometry conceptsin various formats, which may appear onthe standardized tests that they mayencounter. This practice includes multiple-choice, grid-in, and short-responsequestions. Bubble-in and grid-in answersections are provided on the master.

Answers• Page A1 is an answer sheet for the

Standardized Test Practice questionsthat appear in the Student Edition onpages 458–459. This improves students’familiarity with the answer formats theymay encounter in test taking.

• The answers for the lesson-by-lessonmasters are provided as reduced pageswith answers appearing in red.

• Full-size answer keys are provided forthe assessment masters in this booklet.

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

88

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 8.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

diagonals

isosceles trapezoid

kite

median

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

parallelogram

rectangle

rhombus

square

trapezoid

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

88

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

88

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 8. As you

study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 8.1Interior Angle Sum Theorem

Theorem 8.2Exterior Angle Sum Theorem

Theorem 8.3

Theorem 8.4

Theorem 8.5

Theorem 8.7

Theorem 8.8

(continued on the next page)

© Glencoe/McGraw-Hill x Glencoe Geometry

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 8.12

Theorem 8.13

Theorem 8.15

Theorem 8.17

Theorem 8.18

Theorem 8.19

Theorem 8.20

Learning to Read MathematicsProof Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

88

Study Guide and InterventionAngles of Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

© Glencoe/McGraw-Hill 417 Glencoe Geometry

Less

on

8-1

Sum of Measures of Interior Angles The segments that connect thenonconsecutive sides of a polygon are called diagonals. Drawing all of the diagonals fromone vertex of an n-gon separates the polygon into n ! 2 triangles. The sum of the measuresof the interior angles of the polygon can be found by adding the measures of the interiorangles of those n ! 2 triangles.

Interior Angle If a convex polygon has n sides, and S is the sum of the measures of its interior angles, Sum Theorem then S " 180(n ! 2).

A convex polygon has 13 sides. Find the sum of the measuresof the interior angles.S " 180(n ! 2)

" 180(13 ! 2)" 180(11)" 1980

The measure of aninterior angle of a regular polygon is120. Find the number of sides.The number of sides is n, so the sum of themeasures of the interior angles is 120n.

S " 180(n ! 2)120n " 180(n ! 2)120n " 180n ! 360

!60n " !360n " 6

Example 1Example 1 Example 2Example 2

ExercisesExercises

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

1. 10-gon 2. 16-gon 3. 30-gon

4. 8-gon 5. 12-gon 6. 3x-gon

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

7. 150 8. 160 9. 175

10. 165 11. 168.75 12. 135

13. Find x.

7x !

(4x " 10)!

(4x " 5)!

(6x " 10)!

(5x # 5)!

A B

C

D

E


Sum of Measures of Exterior Angles There is a simple relationship among theexterior angles of a convex polygon.

Exterior Angle If a polygon is convex, then the sum of the measures of the exterior angles, Sum Theorem one at each vertex, is 360.

Find the sum of the measures of the exterior angles, one at eachvertex, 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 " 360n " 60

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

1. 10-gon 2. 16-gon 3. 36-gon

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

4. 12-gon 5. 36-gon 6. 2x-gon

Find the measure of an exterior angle given the number of sides of a regularpolygon.

7. 40 8. 18 9. 12

10. 24 11. 180 12. 8

A B

C

DE

F

Study Guide and Intervention (continued)

Angles of Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeAngles of Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1


Less

on

8-1


1. nonagon 2. heptagon 3. decagon


4. 108 5. 120 6. 150

Find the measure of each interior angle using the given information.

7. 8.

9. quadrilateral STUW with !S ! !T, 10. hexagon DEFGHI with !U ! !W, m!S " 2x # 16, !D ! !E ! !G ! !H, !F ! !I,m!U " x # 14 m!D " 7x, m!F " 4x

Find the measures of an interior angle and an exterior angle for each regularpolygon.

11. quadrilateral 12. pentagon 13. dodecagon

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

14. 8 15. 9 16. 13

H G

FI

D ES T

UW

2x !

(2x " 20)! (3x # 10)!

(2x # 10)!

L M

NP

x !

x !

(2x # 15)!

(2x # 15)!

A B

CD



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


4. 144 5. 156 6. 160

Find the measure of each interior angle using the given information.

7. 8. quadrilateral RSTU with m!R " 6x ! 4, m!S " 2x # 8

Find the measures of an interior angle and an exterior angle for each regularpolygon. Round to the nearest tenth if necessary.

9. 16-gon 10. 24-gon 11. 30-gon

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

12. 14 13. 22 14. 40

15. CRYSTALLOGRAPHY Crystals are classified according to seven crystal systems. Thebasis of the classification is the shapes of the faces of the crystal. Turquoise belongs tothe 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.

R S

TU

x !

(2x " 15)! (3x # 20)!

(x " 15)!

J K

MN

Practice Angles of Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Reading to Learn MathematicsAngles of Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1


Less

on

8-1

Pre-Activity How does a scallop shell illustrate the angles of polygons?

Read the introduction to Lesson 8-1 at the top of page 404 in your textbook.

• How many diagonals of the scallop shell shown in your textbook can bedrawn from vertex A?

• How many diagonals can be drawn from one vertex of an n-gon? Explainyour reasoning.

Reading the Lesson1. Write an expression that describes each of the following quantities for a regular n-gon. If

the expression applies to regular polygons only, write regular. If it applies to all convexpolygons, write all.

a. the sum of the measures of the interior angles

b. the measure of each interior angle

c. the sum of the measures of the exterior angles (one at each vertex)

d. the measure of each exterior angle

2. Give the measure of an interior angle and the measure of an exterior angle of each polygon.a. equilateral triangle c. squareb. regular hexagon d. regular octagon

3. Underline the correct word or phrase to form a true statement about regular polygons.a. As the number of sides increases, the sum of the measures of the interior angles

(increases/decreases/stays the same).b. As the number of sides increases, the measure of each interior angle

(increases/decreases/stays the same).c. As the number of sides increases, the sum of the measures of the exterior angles

(increases/decreases/stays the same).d. As the number of sides increases, the measure of each exterior angle

(increases/decreases/stays the same).e. If a regular polygon has more than four sides, each interior angle will be a(n)

(acute/right/obtuse) angle, and each exterior angle will be a(n) (acute/right/obtuse) angle.

Helping You Remember4. A good way to remember a new mathematical idea or formula is to relate it to something

you already know. How can you use your knowledge of the Angle Sum Theorem (for atriangle) to help you remember the Interior Angle Sum Theorem?


TangramsThe tangram puzzle is composed of seven pieces that form a square, as shown at theright. This puzzle has been a popularamusement for Chinese students for hundreds and perhaps thousands of years.

Make a careful tracing of the figure above. Cut out the pieces and rearrange them to form each figure below. Record each answer by drawing lines within each figure.

1. 2.

3. 4.

5. 6.

7. Create a different figure using the seven tangram pieces. Trace the outline.Then challenge another student to solve the puzzle.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-18-1

Study Guide and InterventionParallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2


Less

on

8-2

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

If PQRS is a parallelogram, then

The opposite sides of a P"Q" ! S"R" and P"S" !Q"R"parallelogram are congruent.

The opposite angles of a !P ! !R and !S ! !Qparallelogram are congruent.

The consecutive angles of a !P and !S are supplementary; !S and !R are supplementary; parallelogram are supplementary. !R and !Q are supplementary; !Q and !P are supplementary.

If a parallelogram has one right If m!P " 90, then m!Q " 90, m!R " 90, and m!S " 90.angle, then it has four right angles.

If ABCD is a parallelogram, find a and b.A"B" and C"D" are opposite sides, so A"B" ! C"D".2a " 34a " 17

!A and !C are opposite angles, so !A ! !C.8b " 112b " 14

Find x and y in each parallelogram.

1. 2.

3. 4.

5. 6.

72x

30x 150

2y60!

55! 5x!

2y!

6x! 12x!

3y!3y

12

6x

8y

88

6x!

4y!

3x!

BA

D C34

2a8b!

112!

P Q

RS

ExampleExample

ExercisesExercises


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

If ABCD is a parallelogram, then:

The diagonals of a parallelogram bisect each other. AP " PC and DP " PB

Each diagonal separates a parallelogram "ACD ! "CAB and "ADB ! "CBDinto two congruent triangles.

Find x and y in parallelogram ABCD.The diagonals bisect each other, so AE " CE and DE " BE.6x " 24 4y " 18x " 4 y " 4.5

Find x and y in each parallelogram.

1. 2. 3.

4. 5. 6.

Complete each statement about !ABCD.Justify your answer.

7. !BAC !

8. D"E" !

9. "ADC !

10. A"D" ||

A B

CDE

x

17

4y3x! 2y

12

3x

30!10

y

2x!

60! 4y!

4x

2y28

3x4y

812

A BE

CD

6x

1824

4y

A B

CD

P


Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

ExampleExample

ExercisesExercises

Skills PracticeParallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2


Less

on

8-2

Complete each statement about !DEFG. Justify your answer.

1. D"G" ||

2. D"E" !

3. G"H" !

4. !DEF !

5. !EFG is supplementary to .

6. "DGE !

ALGEBRA Use !WXYZ to find each measure or value.

7. m!XYZ " 8. m!WZY "

9. m!WXY " 10. a "

COORDINATE GEOMETRY Find the coordinates of the intersection of thediagonals of parallelogram HJKL given each set of vertices.

11. H(1, 1), J(2, 3), K(6, 3), L(5, 1) 12. H(!1, 4), J(3, 3), K(3, !2), L(!1, !1)

13. PROOF Write a paragraph proof of the theorem Consecutive angles in a parallelogramare supplementary.

D C

BA

YZ

XWA

50!30

2a

70!

?

?

?

?

?

? FG

ED H


Complete each statement about !LMNP. Justify your answer.

1. L"Q" !

2. !LMN !

3. "LMP !

4. !NPL is supplementary to .

5. L"M" !

ALGEBRA Use !RSTU to find each measure or value.

6. m!RST " 7. m!STU "

8. m!TUR " 9. b "

COORDINATE GEOMETRY Find the coordinates of the intersection of thediagonals of parallelogram PRYZ given each set of vertices.

10. P(2, 5), R(3, 3), Y(!2, !3), Z(!3, !1) 11. P(2, 3), R(1, !2), Y(!5, !7), Z(!4, !2)

12. PROOF Write a paragraph proof of the following.Given: #PRST and #PQVUProve: !V ! !S

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

1 2

34

P R

S

U V

Q

T

TU

SRB30! 23

4b # 1

25!

?

?

?

?

?P N

MQ

L

Practice Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

Reading to Learn MathematicsParallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2


Less

on

8-2

Pre-Activity How are parallelograms used to represent data?


• What is the name of the shape of the top surface of each wedge of cheese?

• Are the three polygons shown in the drawing similar polygons? Explain yourreasoning.

Reading the Lesson

1. Underline words or phrases that can complete the following sentences to make statementsthat are always true. (There may be more than one correct choice for some of the sentences.)

a. Opposite sides of a parallelogram are (congruent/perpendicular/parallel).

b. Consecutive angles of a parallelogram are (complementary/supplementary/congruent).

c. A diagonal of a parallelogram divides the parallelogram into two(acute/right/obtuse/congruent) triangles.

d. Opposite angles of a parallelogram are (complementary/supplementary/congruent).

e. The diagonals of a parallelogram (bisect each other/are perpendicular/are congruent).

f. If a parallelogram has one right angle, then all of its other angles are(acute/right/obtuse) angles.

2. Let ABCD be a parallelogram with AB $ BC and with no right angles.

a. Sketch a parallelogram that matches the descriptionabove and draw diagonal B"D".

In parts b–f, complete each sentence.

b. A"B" || and A"D" || .

c. A"B" ! and B"C" ! .

d. !A ! and !ABC ! .

e. !ADB ! because these two angles are

angles formed by the two parallel lines and and the

transversal .

f. "ABD ! .

Helping You Remember

3. A good way to remember new theorems in geometry is to relate them to theorems youlearned earlier. Name a theorem about parallel lines that can be used to remember thetheorem that says, “If a parallelogram has one right angle, it has four right angles.”

DA

CB


TessellationsA tessellation is a tiling pattern made of polygons. The pattern can be extended so that the polygonal tiles cover the plane completely withno gaps. A checkerboard and a honeycomb pattern are examples oftessellations. Sometimes the same polygon can make more than onetessellation pattern. Both patterns below can be formed from anisosceles triangle.

Draw a tessellation using each polygon.

1. 2. 3.

Draw two different tessellations using each polygon.

4. 5.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-28-2

Study Guide and InterventionTests for Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3


Less

on

8-3

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

If: If:

both pairs of opposite sides are parallel, A"B" || D"C" and A"D" || B"C",

both pairs of opposite sides are congruent, A"B" ! D"C" and A"D" ! B"C",

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

the diagonals bisect each other, A"E" ! C"E" and D"E" ! B"E",

one pair of opposite sides is congruent and parallel, A"B" || C"D" and A"B" ! C"D", or A"D" || B"C" and A"D" !B"C",

then: the figure is a parallelogram. then: ABCD 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 " 26x " 12 4(2) ! 2y " 2x " 2 8 ! 2y " 2

!2y " !6y " 3

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

1. 2.

3. 4.

5. 6. 6y!

3x!

(x " y)!2x!

30!24!

9x!6y

45!185x!5y!

25!

11x!

5y! 55!2x # 2

2y 8

12

J H

GF 6x " 3

154x # 2y 2

A B

CD

E

ExercisesExercises

ExampleExample


Parallelograms on the Coordinate Plane On the coordinate plane, the DistanceFormula and the Slope Formula 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 " %yx

2

2

!

!

yx

1

1%.

slope of A"D" " %!2

3!!

(0!3)

% " %31% " 3 slope of B"C" " %

23!

!(!

21)

% " %31% " 3

slope of A"B" " %3

2!

!(!

32)

% " !%15% slope of C"D" " %

2!

!1

(!!

03)

% " !%15%

Opposite sides have the same slope, so A"B" || C"D" and A"D" || B"C". Both pairs of opposite sidesare parallel, so ABCD is a parallelogram.

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"Both pairs of opposite sides have the same length, so ABCD is a parallelogram.

Determine whether a figure with the given vertices is a parallelogram. Use themethod 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 possiblecoordinates for the fourth vertex.

x

y

O

C

A

D

B


Tests for Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3

ExampleExample

ExercisesExercises

Skills PracticeTests for Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3


Less

on

8-3

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

1. 2.

3. 4.

COORDINATE GEOMETRY Determine whether a figure with the given vertices is aparallelogram. Use 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 Formula

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.

3x # 14

x " 20

3y " 2y " 20

(4x # 35)!

(3x " 10)!(2y # 5)!

(y " 15)!

2y # 11

y " 3 4x # 3

3x

x " 16

y " 192y

2x # 8


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

1. 2.

3. 4.

COORDINATE GEOMETRY Determine whether a figure with the given vertices is aparallelogram. Use 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 Formula

ALGEBRA Find x and y so that each 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 areparallelograms?

#4y # 2

x " 12

#2x " 6

y " 23#4x " 6

#6x

7y " 312y # 7

3y # 5#3x " 4

#4x # 22y " 8(5x " 29)!

(7x # 11)!(3y " 15)!

(5y # 9)!

118! 62!

118!62!

Practice Tests for Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3

Reading to Learn MathematicsTests for Parallelograms

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3


Less

on

8-3

Pre-Activity How are parallelograms used in architecture?


Make two observations about the angles in the roof of the covered bridge.

Reading the Lesson1. Which of the following conditions guarantee that a quadrilateral is a parallelogram?

A. Two sides are parallel.B. Both pairs of opposite sides are congruent.C. The diagonals are perpendicular.D. A pair of opposite sides is both parallel and congruent.E. There are two right angles.F. The sum of the measures of the interior angles is 360.G. All four sides are congruent.H. Both pairs of opposite angles are congruent.I. Two angles are acute and the other two angles are obtuse.J. The diagonals bisect each other.K. The diagonals are congruent.L. All four angles are right angles.

2. Determine whether there is enough given information to know that each figure is aparallelogram. If so, state the definition or theorem that justifies your conclusion.

a. b.

c. d.

Helping You Remember3. A good way to remember a large number of mathematical ideas is to think of them in

groups. How can you state the conditions as one group about the sides of quadrilateralsthat guarantee that the quadrilateral is a parallelogram?


Tests for ParallelogramsBy definition, a quadrilateral is a parallelogram if and only if both pairs of opposite sidesare parallel. What conditions other than both pairs of opposite sides parallel will guaranteethat a quadrilateral is a parallelogram? In this activity, several possibilities will beinvestigated 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?

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-38-3

Study Guide and InterventionRectangles

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4


Less

on

8-4

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. T"R" ! U"S"

T S

RU

Q

In rectangle RSTUabove, US $ 6x " 3 and RT $ 7x # 2.Find x.The diagonals of a rectangle bisect eachother, so US " RT.

6x # 3 " 7x ! 23 " x ! 25 " x

In rectangle RSTUabove, 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 " 9024x ! 6 " 90

24x " 96x " 4

m!STR " 8x # 3 " 8(4) # 3 or 35

Example 1Example 1 Example 2Example 2

ExercisesExercises

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.

9. In rectangle MNOP, m!1 " 40. Find the measure of each numbered angle.

M N

OP

K

1 23

45

6

B C

DA

E


Prove that Parallelograms Are Rectangles The diagonals of a rectangle arecongruent, 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, andproperties of diagonals to show that a figure is a rectangle.

Determine whether A(#3, 0), B(#2, 3), C(4, 1),and D(3, #2) are the vertices of a rectangle.

Method 1: Use the Slope Formula.

slope of A"B" " %!2

3!!

(0!3)

% " %31% or 3 slope of A"D" " %

3!

!2

(!!

03)

% " %!62% or !%

13%

slope of C"D" " %!32!!

41

% " %!!

31% or 3 slope of B"C" " %

41!

!(!

32)

% " %!62% or !%

13%

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

Method 2: Use the Midpoint and Distance Formulas.

The midpoint of A"C" is $%!32# 4%, %

0 #2

1%% " $%

12%, %

12%% and the midpoint of B"D" is

$%!22# 3%, %

3 !2

2%% " $%

12%, %

12%%. The diagonals have the same midpoint so they bisect each other.

Thus, ABCD is a parallelogram.

AC " #(!3 !" 4)2 #" (0 !"1)2" " #49 #"1" or #50"BD " #(!2 !" 3)2 #" (3 !"(!2))2" " #25 #"25" or #50"The diagonals are congruent. ABCD is a parallelogram with diagonals that bisect eachother, so it is a rectangle.

Determine whether ABCD is a rectangle given each set of vertices. Justify youranswer.

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

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

5. A(!1, !5), B(!3, 0), C(2, 2), D(4, !3) 6. A(!1, !1), B(0, 2), C(4, 3), D(3, 0)

7. A parallelogram has vertices R(!3, !1), S(!1, 2), and T(5, !2). Find the coordinates ofU so that RSTU is a rectangle.

x

y

O

D

B

A

E C


Rectangles

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4

ExampleExample

ExercisesExercises

Skills PracticeRectangles

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4


Less

on

8-4

ALGEBRA ABCD is a rectangle.

1. If AC " 2x # 13 and DB " 4x ! 1, find x.

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 x.

6. If m!BDC " 7x # 1 and m!ADB " 9x ! 7, find m!BDC.

7. If m!ABD " x2 ! 7 and m!CDB " 4x # 5, find x.

8. If m!BAC " x2 # 3 and m!CAD " x # 15, find m!BAC.

PRST is a rectangle. Find each measure if m"1 $ 50.

9. m!2 10. m!3

11. m!4 12. m!5

13. m!6 14. m!7

15. m!8 16. m!9

COORDINATE GEOMETRY Determine whether TUXY is a rectangle given each setof vertices. Justify your answer.

17. T(!3, !2), U(!4, 2), X(2, 4), Y(3, 0)

18. T(!6, 3), U(0, 6), X(2, 2), Y(!4, !1)

19. T(4, 1), U(3, !1), X(!3, 2), Y(!2, 4)

T

1 2 3 4

567 89

S

RP

D C

BAE


ALGEBRA 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 " x2 # 9 and m!UTR " 2x # 44, find x.

6. If m!RSU " x2 ! 1 and m!TUS " 3x # 9, find m!RSU.

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 Determine whether BGHL is a rectangle given each setof vertices. Justify your answer.

13. B(!4, 3), G(!2, 4), H(1, !2), L(!1, !3)

14. B(!4, 5), G(6, 0), H(3, !6), L(!7, !1)

15. B(0, 5), G(4, 7), H(5, 4), L(1, 2)

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

K

2 1 5

436 7

J

HG

U T

SRZ

Practice Rectangles

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4

Reading to Learn MathematicsRectangles

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4


Less

on

8-4

Pre-Activity How are rectangles used in tennis?


Are the singles court and doubles court similar rectangles? Explain youranswer.

Reading the Lesson1. Determine whether each sentence is always, sometimes, or never true.

a. If a quadrilateral has four congruent angles, it is a rectangle.

b. If consecutive angles of a quadrilateral are supplementary, then the quadrilateral is a rectangle.

c. The diagonals of a rectangle bisect each other.

d. If the diagonals of a quadrilateral bisect each other, the quadrilateral is a rectangle.

e. Consecutive angles of a rectangle are complementary.

f. Consecutive angles of a rectangle are congruent.

g. If the diagonals of a quadrilateral are congruent, the quadrilateral is a rectangle.

h. A diagonal of a rectangle bisects two of its angles.

i. A diagonal of a rectangle divides the rectangle into two congruent right triangles.

j. If the diagonals of a quadrilateral bisect each other and are congruent, thequadrilateral is a rectangle.

k. If a parallelogram has one right angle, it is a rectangle.

l. If a parallelogram has four congruent sides, it is a rectangle.

2. ABCD is a rectangle with AD & AB.Name each of the following in this figure.

a. all segments that are congruent to B"E"b. all angles congruent to !1

c. all angles congruent to !7

d. two pairs of congruent triangles

Helping You Remember3. It is easier to remember a large number of geometric relationships and theorems if you

are able to combine some of them. How can you combine the two theorems aboutdiagonals that you studied in this lesson?

A1

2 3 4 5

678D

CBE


Counting Squares and RectanglesEach puzzle below contains many squares and/or rectangles.Count them carefully. You may want to label each region so youcan list all possibilities.

How many rectangles are in the figure at the right?

Label each small region with a letter. Then list each rectangle by writing the letters of regions it contains.

A, B, C, D, AB, CD, AC, BD, ABCD

There are 9 rectangles.

How many squares are in each figure?

1. 2. 3.

How many rectangles are in each figure?

1. 2. 3.

A B

C D

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-48-4

ExampleExample

Study Guide and InterventionRhombi and Squares

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5


Less

on

8-5

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

The diagonals are perpendicular. M"H" ⊥ R"O"

Each diagonal bisects a pair of opposite angles. M"H" bisects !RMO and !RHO.R"O" bisects !MRH and !MOH.

If the diagonals of a parallelogram are If RHOM is a parallelogram and R"O" ⊥ M"H",perpendicular, then the figure is a rhombus. then RHOM is a rhombus.

In rhombus ABCD, m"BAC $ 32. Find the measure of each numbered angle.ABCD is a rhombus, so the diagonals are perpendicular and "ABEis a right triangle. Thus m!4 " 90 and m!1 " 90 ! 32 or 58. Thediagonals 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 arealternate interior angles for parallel lines, so m!3 " 32.

ABCD is a rhombus.

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.

9. a. What is the midpoint of F"H"?

b. What is the midpoint of G"J"?

c. What kind of figure is FGHJ? Explain.

d. What is the slope of F"H"?

e. What is the slope of G"J"?

f. Based on parts c, d, and e, what kind of figure is FGHJ ? Explain.

x

y

O

J

GF

H

C

A D

EB

CA

D

E

B

32!1 2

34

M O

HRB

ExampleExample

ExercisesExercises


Properties of Squares A square has all the properties of a rhombus and all theproperties of a rectangle.

Find the measure of each numbered angle of square ABCD.

Using properties of rhombi and rectangles, the diagonals are perpendicular and congruent. "ABE is a right triangle, so m!1 " m!2 " 90.

Each vertex angle is a right angle and the diagonals bisect the vertex angles,so m!3 " m!4 " m!5 " 45.

Determine whether the given vertices represent a parallelogram, rectangle,rhombus, or square. Explain your reasoning.

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

2. D(!2, 1), E(!1, 3), F(3, 1), G(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. S(!1, 4), T(3, 2), U(1, !2), V(!3, 0)

6. F(!1, 0), G(1, 3), H(4, 1), I(2, !2)

7. Square RSTU has vertices R(!3, !1), S(!1, 2), and T(2, 0). Find the coordinates ofvertex U.

C

A D

1 2345

EB


Rhombi and Squares

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5

ExercisesExercises

ExampleExample

Skills PracticeRhombi and Squares

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5


Less

on

8-5

Use rhombus DKLM with AM $ 4x, AK $ 5x # 3, and DL $ 10.

1. Find x. 2. Find AL.

3. Find m!KAL. 4. Find DM.

Use rhombus RSTV with RS $ 5y " 2, ST $ 3y " 6, and NV $ 6.

5. Find y. 6. Find TV.

7. Find m!NTV. 8. Find m!SVT.

9. Find m!RST. 10. Find m!SRV.

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

11. Q(3, 5), R(3, 1), S(!1, 1), T(!1, 5)

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

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

14. Q(2, !4), R(!6, !8), S(!10, 2), T(!2, 6)

R V

NS T

M L

AD K


Use rhombus PRYZ with RK $ 4y " 1, ZK $ 7y # 14,PK $ 3x # 1, and YK $ 2x " 6.

1. Find PY. 2. Find RZ.

3. Find RY. 4. Find m!YKZ.

Use rhombus MNPQ with PQ $ 3!2", PA $ 4x # 1, and AM $ 9x # 6.

5. Find AQ. 6. Find m!APQ.

7. Find m!MNP. 8. Find PM.

COORDINATE GEOMETRY Given each set of vertices, determine whether !BEFGis a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning.

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(!7, !1), G(!2, !1)

12. TESSELATIONS 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.

AN

M Q

P

K

P

YR

Z

Practice Rhombi and Squares

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5

Reading to Learn MathematicsRhombi and Squares

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5


Less

on

8-5

Pre-Activity How can you ride a bicycle with square wheels?


If you draw a diagonal on the surface of one of the square wheels shown inthe picture in your textbook, how can you describe the two triangles that areformed?

Reading the Lesson

1. Sketch each of the following.

a. a quadrilateral with perpendicular b. a quadrilateral with congruent diagonals that is not a rhombus diagonals that is not a rectangle

c. a quadrilateral whose diagonals are perpendicular and bisect each other,but are not congruent

2. List all of the following special quadrilaterals that have each listed property:parallelogram, rectangle, rhombus, square.

a. The diagonals are congruent.

b. Opposite sides are congruent.

c. The diagonals are perpendicular.

d. Consecutive angles are supplementary.

e. The quadrilateral is equilateral.

f. The quadrilateral is equiangular.

g. The diagonals are perpendicular and congruent.

h. A pair of opposite sides is both parallel and congruent.

3. What is the common name for a regular quadrilateral? Explain your answer.


4. A good way to remember something is to explain it to someone else. Suppose that yourclassmate Luis is having trouble remembering which of the properties he has learned inthis chapter apply to squares. How can you help him?


Creating Pythagorean PuzzlesBy drawing two squares and cutting them in a certain way, youcan make a puzzle that demonstrates the Pythagorean Theorem.A sample puzzle is shown. You can create your own puzzle byfollowing 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 # b"2".

a

a

b X

b

b

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-58-5

Study Guide and InterventionTrapezoids

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6


Less

on

8-6

Properties of Trapezoids A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called basesand the nonparallel sides are called legs. If the legs are congruent,the trapezoid is an isosceles trapezoid. In an isosceles trapezoid both pairs of base angles are congruent. STUR is an isosceles trapezoid.

S"R" ! T"U"; !R ! !U, !S !!T

The vertices of ABCD are A(#3, #1), B(#1, 3),C(2, 3), and D(4, #1). Verify that ABCD is a trapezoid.

slope of A"B" " %!

31!

!

(!(!

13))

% " %42% " 2

slope of A"D" " %!

41!

!

(!(!

31))

% " %07% " 0

slope of B"C" " %2

3!

!(!

31)

% " %03% " 0

slope of C"D" " %!41!!

23

% " %!24% " !2

Exactly two sides are parallel, A"D" and B"C", so ABCD is a trapezoid. AB " CD, so ABCD isan isosceles trapezoid.

In Exercises 1–3, determine whether ABCD is a trapezoid. If so, determinewhether it is an isosceles trapezoid. Explain.

1. A(!1, 1), B(2, 1), C(3, !2), and D(2, !2)

2. A(3, !3), B(!3, !3), C(!2, 3), and D(2, 3)

3. A(1, !4), B(!3, !3), C(!2, 3), and D(2, 2)

4. The vertices of an isosceles trapezoid are R(!2, 2), S(2, 2), T(4, !1), and U(!4, !1).Verify that the diagonals are congruent.

x

y

O

B

DA

C

T

R U

base

base

Sleg

AB " #(!3 !" (!1))"2 # (!"1 ! 3")2"" #4 # 1"6" " #20" " 2#5"

CD " #(2 ! 4")2 # ("3 ! (!"1))2"" #4 # 1"6" " #20" " 2#5"

ExercisesExercises

ExampleExample


Medians of Trapezoids The median of a trapezoid is the segment that joins the midpoints of the legs. It is parallel to the bases, and its length is one-half the sum of the lengths of the bases.

In trapezoid HJKL, MN " %12%(HJ # LK ).

M"N" is the median of trapezoid RSTU. Find x.

MN " %12%(RS # UT)

30 " %12%(3x # 5 # 9x ! 5)

30 " %12%(12x)

30 " 6x5 " x

M"N" is the median of trapezoid HJKL. Find each indicated value.

1. Find MN if HJ " 32 and LK " 60.

2. Find LK if HJ " 18 and MN " 28.

3. Find MN if HJ # LK " 42.

4. Find m!LMN if m!LHJ " 116.

5. Find m!JKL if HJKL is isosceles and m!HLK " 62.

6. Find HJ if MN " 5x # 6, HJ " 3x # 6, and LK " 8x.

7. Find the length of the median of a trapezoid with vertices A(!2, 2), B(3, 3), C(7, 0), and D(!3, !2).

L K

NM

H J

U T

NM

R 3x " 530

9x # 5

S

L K

NM

H J


Trapezoids

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6

ExampleExample

ExercisesExercises

Skills PracticeTrapezoids

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6


Less

on

8-6

COORDINATE GEOMETRY ABCD is a quadrilateral with vertices A(#4, #3),B(3, #3), C(6, 4), D(#7, 4).

1. Verify that ABCD is a trapezoid.

2. Determine whether ABCD is an isosceles trapezoid. Explain.

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

3. Verify that EFGH is a trapezoid.

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

COORDINATE GEOMETRY LMNP is a quadrilateral with vertices L(#1, 3), M(#4, 1),N(#6, 3), P(0, 7).

5. Verify that LMNP is a trapezoid.

6. Determine whether LMNP is an isosceles trapezoid. Explain.

ALGEBRA Find the missing measure(s) for the given trapezoid.

7. For trapezoid HJKL, S and T are 8. For trapezoid WXYZ, P and Q aremidpoints of the legs. Find HJ. midpoints of the legs. Find WX.

9. For trapezoid DEFG, T and U are 10. For isosceles trapezoid QRST, find themidpoints of the legs. Find TU, m!E, length of the median, m!Q, and m!S.and m!G.

ST

RQ125!

60

25

D E

FG

T U85!

35!42

14

W X

YZ

P Q12

19

H J

KL

S T72

86


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

1. Verify that RSTU is a trapezoid.

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

COORDINATE GEOMETRY BGHJ is a quadrilateral with vertices B(#9, 1), G(2, 3),H(12, #2), J(#10, #6).

3. Verify that BGHJ is a trapezoid.

4. Determine whether BGHJ is an isosceles trapezoid. Explain.

ALGEBRA Find the missing measure(s) for the given trapezoid.

5. For trapezoid CDEF, V and Y are 6. For trapezoid WRLP, B and C aremidpoints of the legs. Find CD. midpoints of the legs. Find LP.

7. For trapezoid FGHI, K and M are 8. For isosceles trapezoid TVZY, find the midpoints of the legs. Find FI, m!F, length of the median, m!T, and m!Z.and m!I.

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

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

V

Y Z

T 60!34

5

H

K M

G

IF

140! 125!21

36

W R

LP

B C42

66F E

DC

V Y28

18

Practice Trapezoids

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6

Reading to Learn MathematicsTrapezoids

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6


Less

on

8-6

Pre-Activity How are trapezoids used in architecture?


How might trapezoids be used in the interior design of a home?

Reading the Lesson

1. In the figure at the right, EFGH is a trapezoid, I is the midpoint of F"E", and J is the midpoint of G"H". Identify each of the following segments or angles in the figure.

a. the bases of trapezoid EFGH

b. the two pairs of base angles of trapezoid EFGH

c. the legs of trapezoid EFGH

d. the median of trapezoid EFGH

2. Determine whether each statement is true or false. If the statement is false, explain why.

a. A trapezoid is a special kind of parallelogram.

b. The diagonals of a trapezoid are congruent.

c. The median of a trapezoid is parallel to the legs.

d. The length of the median of a trapezoid is the average of the length of the bases.

e. A trapezoid has three medians.

f. The bases of an isosceles trapezoid are congruent.

g. An isosceles trapezoid has two pairs of congruent angles.

h. The median of an isosceles trapezoid divides the trapezoid into two smaller isoscelestrapezoids.


3. A good way to remember a new geometric theorem is to relate it to one you alreadyknow. Name and state in words a theorem about triangles that is similar to the theoremin this lesson about the median of a trapezoid.

HE

JIGF


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?

10 in.

12 in.

20 in.

60 in.

81 in.

9 in.

3 in.

3 in.

3 in.

9 in.

9 in.

jack rafters

hip rafter

common rafters

Roof Frame

ridge boardvalley rafter

plate

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-68-6

Study Guide and InterventionCoordinate Proof with Quadrilaterals

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7


Less

on

8-7

Position Figures Coordinate proofs use properties of lines and segments to provegeometric properties. The first step in writing a coordinate proof is to place the figure on thecoordinate plane in a convenient way. Use the following guidelines for placing a figure onthe coordinate plane.

1. Use the origin as a vertex, so one set of coordinates is (0, 0), or use the origin as the center of the figure.

2. Place at least one side of the quadrilateral on an axis so you will have some zero coordinates.

3. Try to keep the quadrilateral in the first quadrant so you will have positive coordinates.

4. Use coordinates that make the computations as easy as possible. For example, use even numbers if you are going to be finding midpoints.

Position and label a rectangle with sides aand b units long on the coordinate plane.

• Place one vertex at the origin for R, so one vertex is R(0, 0).• Place side R"U" along the x-axis and side R"S" along the y-axis,

with the rectangle in the first quadrant.• The sides are a and b units, so label two vertices S(0, a) and

U(b, 0).• Vertex T is b units right and a units up, so the fourth vertex is T(b, a).

Name the missing coordinates for each quadrilateral.

1. 2. 3.

Position and label each quadrilateral on the coordinate plane.

4. square STUV with 5. parallelogram PQRS 6. rectangle ABCD withside s units with congruent diagonals length twice the width

x

y

D(2a, 0)

C(2a, a)B(0, a)

A(0, 0)x

y

S(a, 0)

R(a, b)Q(0, b)

P(0, 0)x

y

V(s, 0)

U(s, s)T(0, s)

S(0, 0)

x

y

D(a, 0)

C(?, c)B(b, ?)

A(0, 0)x

y

Q(a, 0)

P(a # b, c)N(?, ?)

M(0, 0)x

y

B(c, 0)

C(?, ?)D(0, c)

A(0, 0)

x

y

U(b, 0)

T(b, a)S(0, a)

R(0, 0)

ExercisesExercises

ExampleExample


Prove Theorems After a figure has been placed on the coordinate plane and labeled, acoordinate proof can be used to prove a theorem or verify a property. The Distance Formula,the Slope Formula, and the Midpoint Theorem are often used in a coordinate proof.

Write a coordinate proof to show that the diagonals of a square are perpendicular.The first step is to position and label a square on the coordinate plane. Place it in the first quadrant, with one side on each axis.Label the vertices and draw the diagonals.

Given: square RSTUProve: S"U" ⊥ R"T"Proof: The slope of S"U" is %0a

!!

a0% " !1, and the slope of R"T" is %aa

!!

00% " 1.

The product of the two slopes is !1, so S"U" ⊥ R"T".

Write a coordinate proof to show that the length of the median of a trapezoid ishalf the sum of the lengths of the bases.

D(2a, 0)

B(2b, 2c)

M N

C(2d, 2c)

A(0, 0)

x

y

U(a, 0)

T(a, a)S(0, a)

R(0, 0)


Coordinate Proof With Quadrilaterals

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7

ExampleExample

ExerciseExercise

Skills PracticeCoordinate Proof with Quadrilaterals

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7


Less

on

8-7


1. rectangle with length 2a units and 2. isosceles trapezoid with height a units,height a units bases c ! b units and b # c units


3. rectangle 4. rectangle

5. parallelogram 6. isosceles trapezoid

Position and label the figure on the coordinate plane. Then write a coordinateproof for the following.

7. The segments joining the midpoints of the sides of a rhombus form a rectangle.

B(0, 2b)

D(0, #2b)P

NM

Q

C(2a, 0)

A(#2a, 0)

x

y

T(b, 0)

W(?, c)Y(?, ?)

S(a, 0)Ox

y

S(a, ?)

R(?, ?)P(b, c)

T(0, 0)O

x

y

E(5b, 0)

F(5b, 2b)G(?, ?)

D(0, 0)Ox

y

U(c, 0)

C(?, ?)D(0, a)

A(0, 0)O

B(b " c, 0)

B(b, a) C(c, a)

A(0, 0)B(2a, 0)

D(0, a) C(2a, a)

A(0, 0)



1. parallelogram with side length b units 2. isosceles trapezoid with height b units,and height a units bases 2c ! a units and 2c # a units


3. parallelogram 4. isosceles trapezoid

Position and label the figure on the coordinate plane. Then write a coordinateproof for the following.

5. The opposite sides of a parallelogram are congruent.

6. THEATER A stage is in the shape of a trapezoid. Write a coordinate proof to prove that T"R" and S"F" are parallel.

x

y

T(10, 0)

S(30, 25)F(0, 25)

R(20, 0)O

B(a, 0)

D(b, c) C(a " b, c)

A(0, 0)

x

y

W(a, 0)

Z(b, c)Y(?, ?)

X(?, ?) Ox

y

J(2b, 0)

K(?, a)L(c, ?)

H(0, 0)O

B(a " 2c, 0)

D(a, b) C(2c, b)

A(0, 0)B(b, 0)

D(c, a) C(b " c, a)

A(0, 0)

Practice Coordinate Proof with Quadrilaterals

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7

Reading to Learn MathematicsCoordinate Proof with Quadrilaterals

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7


Less

on

8-7

Pre-Activity How can you use a coordinate plane to prove theorems aboutquadrilaterals?


What special kinds of quadrilaterals can be placed on a coordinate system sothat two sides of the quadrilateral lie along the axes?

Reading the Lesson1. Find the missing coordinates in each figure. Then write the coordinates of the four

vertices of the quadrilateral.

a. isosceles trapezoid b. parallelogram

2. Refer to quadrilateral EFGH.

a. Find the slope of each side.

b. Find the length of each side.

c. Find the slope of each diagonal.

d. Find the length of each diagonal.

e. What can you conclude about the sides of EFGH?

f. What can you conclude about the diagonals of EFGH?

g. Classify EFGH as a parallelogram, a rhombus, or a square. Choose the most specificterm. Explain how your results from parts a-f support your conclusion.

Helping You Remember3. What is an easy way to remember how best to draw a diagram that will help you devise

a coordinate proof?

x

y

O

E

FG

H

x

y

Q(b, ?)

P(?, c " a)

N(?, a)

M(?, ?)Ox

y

U(a, ?)

T(b, ?)S(?, c)

R(?, ?) O


Coordinate ProofsAn important part of planning a coordinate proof is correctly placing and labeling thegeometric figure on the coordinate plane.

Draw a diagram you would use in a coordinate proof of thefollowing theorem.

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

In the diagram, note that one vertex, A, is placed at the origin. Also,the coordinates of B, C, and D use 2a, 2b, and 2c in order to avoid the use of fractions when finding the coordinates of the midpoints,M and N.

When doing coordinate proofs, the following strategies may be helpful.

1. If you are asked to prove that segments are parallel or perpendicular, use slopes.

2. If you are asked to prove that segments are congruent or have related measures,use the distance formula.

3. If you are asked to prove that a segment is bisected, use the midpoint formula.

For each of the following theorems, a diagram has been provided to be used in aformal proof. Name the missing coordinates in the diagram. Then, using the Givenand the Prove statements, prove the theorem.

1. The median of a trapezoid is parallel to the bases of the trapezoid. (Use the diagramgiven in the example above.)

Coordinates of M and N:

Given: Trapezoid ABCD has median M"N".Prove: B"C" || M"N" and A"D" || M"N"

Proof:

2. The medians to the legs of an isosceles triangle are congruent.

Coordinates of T and K:

Given: "ABC is isosceles with medians T"B" and K"A".Prove: T"B" ! K"A"

Proof:x

y

O

T K

A(0, 0) B(4a, 0)

C(2a, 2b)

x

y

O

M N

A(0, 0) D(2a, 0)

C(2d, 2c)B(2b, 2c)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

8-78-7

ExampleExample

© Glencoe/McGraw-Hill A2 Glencoe Geometry

Stu

dy

Gu

ide

and I

nte

rven

tion

Ang

les

of P

olyg

ons

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-1

8-1

©G

lenc

oe/M

cGra

w-H

ill41

7G

lenc

oe G

eom

etry

Lesson 8-1

Sum

of

Mea

sure

s o

f In

teri

or

An

gle

sT

he s

egm

ents

tha

t co

nnec

t th

eno

ncon

secu

tive

sid

es o

f a

poly

gon

are

calle

d d

iago

nal

s.D

raw

ing

all o

f th

e di

agon

als

from

one

vert

ex o

f an

n-g

onse

para

tes

the

poly

gon

into

n!

2 tr

iang

les.

The

sum

of

the

mea

sure

sof

the

inte

rior

ang

les

of t

he p

olyg

on c

an b

e fo

und

by a

ddin

g th

e m

easu

res

of t

he in

teri

oran

gles

of

thos

e n

!2

tria

ngle

s.

Inte

rior

Ang

leIf

a co

nvex

pol

ygon

has

nsi

des,

and

Sis

the

sum

of t

he m

easu

res

of it

s in

terio

r an

gles

, S

um T

heor

emth

en S

"18

0(n

!2)

.

A c

onve

x p

olyg

on h

as

13 s

ides

.Fin

d t

he

sum

of

the

mea

sure

sof

th

e in

teri

or a

ngl

es.

S"

180(

n!

2)"

180(

13 !

2)"

180(

11)

"19

80

Th

e m

easu

re o

f an

inte

rior

an

gle

of a

reg

ula

r p

olyg

on i

s12

0.F

ind

th

e n

um

ber

of s

ides

.T

he n

umbe

r of

sid

es is

n,s

o th

e su

m o

f th

em

easu

res

of t

he in

teri

or a

ngle

s is

120

n.S

"18

0(n

!2)

120n

"18

0(n

!2)

120n

"18

0n!

360

!60

n"

!36

0n

"6

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

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.

1.10

-gon

2.16

-gon

3.30

-gon

1440

2520

5040

4.8-

gon

5.12

-gon

6.3x

-gon

1080

1800

180(

3x!

2)

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 s

ides

in

eac

h p

olyg

on.

7.15

08.

160

9.17

512

1872

10.1

6511

.168

.75

12.1

3524

328

13.F

ind

x.20

7x"

( 4x

# 1

0)"

( 4x

# 5

) " ( 6x

# 1

0)"

( 5x

! 5

) "

AB

C

D

E

©G

lenc

oe/M

cGra

w-H

ill41

8G

lenc

oe G

eom

etry

Sum

of

Mea

sure

s o

f Ex

teri

or

An

gle

sT

here

is a

sim

ple

rela

tion

ship

am

ong

the

exte

rior

ang

les

of a

con

vex

poly

gon.

Ext

erio

r A

ngle

If a

poly

gon

is c

onve

x, th

en th

e su

m o

f the

mea

sure

s of

the

exte

rior

angl

es,

Sum

The

orem

one

at e

ach

vert

ex, i

s 36

0.

Fin

d t

he

sum

of

the

mea

sure

s of

th

e ex

teri

or a

ngl

es,o

ne

at e

ach

vert

ex,o

f a

con

vex

27-g

on.

For

any

conv

ex p

olyg

on,t

he s

um o

f th

e m

easu

res

of it

s ex

teri

or a

ngle

s,on

e at

eac

h ve

rtex

,is

360

.

Fin

d t

he

mea

sure

of

each

ext

erio

r an

gle

of

regu

lar

hex

agon

AB

CD

EF

.T

he s

um o

f th

e m

easu

res

of t

he e

xter

ior

angl

es is

360

and

a h

exag

on

has

6 an

gles

.If n

is t

he m

easu

re o

f ea

ch e

xter

ior

angl

e,th

en

6n"

360

n"

60

Fin

d t

he

sum

of

the

mea

sure

s of

th

e ex

teri

or a

ngl

es o

f ea

ch c

onve

x p

olyg

on.

1.10

-gon

2.16

-gon

3.36

-gon

360

360

360

Fin

d t

he

mea

sure

of

an e

xter

ior

angl

e fo

r ea

ch c

onve

x re

gula

r p

olyg

on.

4.12

-gon

5.36

-gon

6.2x

-gon

3010

$18 x0 $

Fin

d t

he

mea

sure

of

an e

xter

ior

angl

e gi

ven

th

e n

um

ber

of s

ides

of

a re

gula

rp

olyg

on.

7.40

8.18

9.12

920

30

10.2

411

.180

12.8

152

45

AB

C

DE

F

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Ang

les

of P

olyg

ons

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-1

8-1

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Answers (Lesson 8-1)


An

swer

s

Skil

ls P

ract

ice

Ang

les

of P

olyg

ons

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-1

8-1

©G

lenc

oe/M

cGra

w-H

ill41

9G

lenc

oe G

eom

etry

Lesson 8-1

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.no

nago

n2.

hept

agon

3.de

cago

n12

6090

014

40

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 s

ides

in

eac

h p

olyg

on.

4.10

85.

120

6.15

05

612

Fin

d t

he

mea

sure

of

each

in

teri

or a

ngl

e u

sin

g th

e gi

ven

in

form

atio

n.

7.8.

m!

A%

115,

m!

B%

65,

m!

L%

100,

m!

M%

110,

m!

C%

115,

m!

D%

65m

!N

%70

,m!

P%

80

9.qu

adri

late

ral S

TU

Ww

ith

!S

!!

T,

10.h

exag

on D

EF

GH

Iw

ith

!U

!!

W,m

!S

"2x

#16

,!

D!

!E

! !

G!

!H

,!F

!!

I,m

!U

"x

#14

m!

D"

7x,m

!F

"4x

m!

S%

116,

m!

T%

116,

m!

D%

140,

m!

E%

140,

m!

U%

64,m

!W

%64

m!

F%

80,m

!G

%14

0,m

!H

%14

0,m

!I%

80

Fin

d t

he

mea

sure

s of

an

in

teri

or a

ngl

e an

d a

n e

xter

ior

angl

e fo

r ea

ch r

egu

lar

pol

ygon

.

11.q

uadr

ilate

ral

12.p

enta

gon

13.d

odec

agon

90,9

010

8,72

150,

30

Fin

d t

he

mea

sure

s of

an

in

teri

or a

ngl

e an

d a

n e

xter

ior

angl

e gi

ven

th

e n

um

ber

ofsi

des

of

each

reg

ula

r p

olyg

on.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

14.8

15.9

16.1

313

5,45

140,

4015

2.3,

27.7

HG

FI

DE

ST

UW

2x"( 2x

# 2

0)"

( 3x

! 1

0)"

( 2x

! 1

0)"

LM

NP

x"

x"

( 2x

! 1

5)"

( 2x

! 1

5)"

AB

CD

©G

lenc

oe/M

cGra

w-H

ill42

0G

lenc

oe G

eom

etry

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.

1.11

-gon

2.14

-gon

3.17

-gon

1620

2160

2700

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 s

ides

in

eac

h p

olyg

on.

4.14

45.

156

6.16

010

1518

Fin

d t

he

mea

sure

of

each

in

teri

or a

ngl

e u

sin

g th

e gi

ven

in

form

atio

n.

7.8.

quad

rila

tera

l RS

TU

wit

h m

!R

"6x

!4,

m!

S"

2x#

8

m!

J%

115,

m!

K%

130,

m!

M%

50,m

!N

%65

m!

R%

128,

m!

S%

52,

m!

T%

128,

m!

U%

52

Fin

d t

he

mea

sure

s of

an

in

teri

or a

ngl

e an

d a

n e

xter

ior

angl

e fo

r ea

ch r

egu

lar

pol

ygon

.Rou

nd

to

the

nea

rest

ten

th i

f n

eces

sary

.

9.16

-gon

10.2

4-go

n11

.30-

gon

157.

5,22

.516

5,15

168,

12

Fin

d t

he

mea

sure

s of

an

in

teri

or a

ngl

e an

d a

n e

xter

ior

angl

e gi

ven

th

e n

um

ber

ofsi

des

of

each

reg

ula

r p

olyg

on.R

oun

d t

o th

e n

eare

st t

enth

if

nec

essa

ry.

12.1

413

.22

14.4

015

4.3,

25.7

163.

6,16

.417

1,9

15.C

RYST

ALL

OG

RA

PHY

Cry

stal

s ar

e cl

assi

fied

acc

ordi

ng t

o se

ven

crys

tal s

yste

ms.

The

basi

s of

the

cla

ssif

icat

ion

is t

he s

hape

s of

the

fac

es o

f th

e cr

ysta

l.T

urqu

oise

bel

ongs

to

the

tric

linic

sys

tem

.Eac

h of

the

six

fac

es o

f tu

rquo

ise

is in

the

sha

pe o

f a

para

llelo

gram

.F

ind

the

sum

of

the

mea

sure

s of

the

inte

rior

ang

les

of o

ne s

uch

face

.36

0

RS

TU

x"

( 2x

# 1

5)"

( 3x

! 2

0)"

( x #

15)

"

JK

MN

Pra

ctic

e (A

vera

ge)

Ang

les

of P

olyg

ons

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-1

8-1



Rea

din

g t

o L

earn

Math

emati

csA

ngle

s of

Pol

ygon

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-1

8-1

©G

lenc

oe/M

cGra

w-H

ill42

1G

lenc

oe G

eom

etry

Lesson 8-1

Pre-

Act

ivit

yH

ow d

oes

a sc

allo

p s

hel

l il

lust

rate

th

e an

gles

of

pol

ygon

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

1 at

the

top

of

page

404

in y

our

text

book

.

•H

ow m

any

diag

onal

s of

the

sca

llop

shel

l sho

wn

in y

our

text

book

can

be

draw

n fr

om v

erte

x A

?9

•H

ow m

any

diag

onal

s ca

n be

dra

wn

from

one

ver

tex

of a

n n-

gon?

Exp

lain

your

rea

soni

ng.

n!

3;S

ampl

e an

swer

:An

n-go

n ha

s n

vert

ices

,bu

t di

agon

als

cann

ot b

e dr

awn

from

a v

erte

x to

itse

lf or

to

eith

er o

f th

e tw

o ad

jace

nt v

ertic

es.T

here

fore

,the

num

ber

ofdi

agon

als

from

one

ver

tex

is 3

less

tha

n th

e nu

mbe

r of

vert

ices

.

Rea

din

g t

he

Less

on

1.W

rite

an

expr

essi

on t

hat

desc

ribe

s ea

ch o

f th

e fo

llow

ing

quan

titi

es f

or a

reg

ular

n-g

on.I

fth

e ex

pres

sion

app

lies

to r

egul

ar p

olyg

ons

only

,wri

te r

egul

ar.I

f it

app

lies

to a

ll co

nvex

poly

gons

,wri

te a

ll.

a.th

e su

m o

f th

e m

easu

res

of t

he in

teri

or a

ngle

s18

0(n

!2)

;all

b.th

e m

easu

re o

f ea

ch in

teri

or a

ngle$18

0(n n

!2)

$;r

egul

arc.

the

sum

of

the

mea

sure

s of

the

ext

erio

r an

gles

(on

e at

eac

h ve

rtex

)36

0;al

ld.

the

mea

sure

of

each

ext

erio

r an

gle

$3 n60 $;r

egul

ar

2.G

ive

the

mea

sure

of a

n in

teri

or a

ngle

and

the

mea

sure

of a

n ex

teri

or a

ngle

of e

ach

poly

gon.

a.eq

uila

tera

l tri

angl

e60

;120

c.sq

uare

90;9

0b.

regu

lar

hexa

gon

120;

60d.

regu

lar

octa

gon

135;

45

3.U

nder

line

the

corr

ect

wor

d or

phr

ase

to f

orm

a t

rue

stat

emen

t ab

out

regu

lar

poly

gons

.a.

As

the

num

ber

of s

ides

incr

ease

s,th

e su

m o

f th

e m

easu

res

of t

he in

teri

or a

ngle

s(i

ncre

ases

/dec

reas

es/s

tays

the

sam

e).

b.A

s th

e nu

mbe

r of

sid

es in

crea

ses,

the

mea

sure

of

each

inte

rior

ang

le(i

ncre

ases

/dec

reas

es/s

tays

the

sam

e).

c.A

s th

e nu

mbe

r of

sid

es in

crea

ses,

the

sum

of

the

mea

sure

s of

the

ext

erio

r an

gles

(inc

reas

es/d

ecre

ases

/sta

ys t

he s

ame)

.d.

As

the

num

ber

of s

ides

incr

ease

s,th

e m

easu

re o

f ea

ch e

xter

ior

angl

e(i

ncre

ases

/dec

reas

es/s

tays

the

sam

e).

e.If

a r

egul

ar p

olyg

on h

as m

ore

than

fou

r si

des,

each

inte

rior

ang

le w

ill b

e a(

n)(a

cute

/rig

ht/o

btus

e) a

ngle

,and

eac

h ex

teri

or a

ngle

will

be

a(n)

(acu

te/r

ight

/obt

use)

ang

le.

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al id

ea o

r fo

rmul

a is

to

rela

te it

to

som

ethi

ngyo

u al

read

y kn

ow.H

ow c

an y

ou u

se y

our

know

ledg

e of

the

Ang

le S

um T

heor

em (

for

atr

iang

le)

to h

elp

you

rem

embe

r th

e In

teri

or A

ngle

Sum

The

orem

?S

ampl

e an

swer

:Th

e su

m o

f th

e m

easu

res

of t

he (

inte

rior

) an

gles

of

a tr

iang

le is

180

.E

ach

time

anot

her

side

is a

dded

,the

pol

ygon

can

be

subd

ivid

ed in

to o

nem

ore

tria

ngle

,so

180

is a

dded

to

the

inte

rior

ang

le s

um.

©G

lenc

oe/M

cGra

w-H

ill42

2G

lenc

oe G

eom

etry

Tang

ram

sT

he t

angr

am p

uzzl

e is

com

pose

d of

sev

en

piec

es t

hat

form

a s

quar

e,as

sho

wn

at t

heri

ght.

Thi

s pu

zzle

has

bee

n a

popu

lar

amus

emen

t fo

r C

hine

se s

tude

nts

for

hund

reds

and

per

haps

tho

usan

ds o

f ye

ars.

Mak

e a

care

ful

trac

ing

of t

he

figu

re a

bove

.Cu

t ou

t th

e p

iece

s an

d r

earr

ange

th

em t

o fo

rm e

ach

fig

ure

bel

ow.R

ecor

d e

ach

an

swer

by

dra

win

g li

nes

wit

hin

ea

ch f

igu

re.

1.2.

3.4.

5.6.

7.C

reat

e a

diff

eren

t fi

gure

usi

ng t

he s

even

tan

gram

pie

ces.

Tra

ce t

he o

utlin

e.T

hen

chal

leng

e an

othe

r st

uden

t to

sol

ve t

he p

uzzl

e.S

ee s

tude

nts’

wor

k.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-1

8-1



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Para

llelo

gram

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-2

8-2

©G

lenc

oe/M

cGra

w-H

ill42

3G

lenc

oe G

eom

etry

Lesson 8-2

Sid

es a

nd

An

gle

s o

f Pa

ralle

log

ram

sA

qua

drila

tera

l wit

h bo

th p

airs

of

oppo

site

sid

es p

aral

lel i

s a

par

alle

logr

am.H

ere

are

four

impo

rtan

t pr

oper

ties

of

para

llelo

gram

s.

If P

QR

Sis

a p

aral

lelo

gram

,the

n

The

opp

osite

sid

es o

f a

P"Q"!

S"R"

and

P"S"!

Q"R"

para

llelo

gram

are

con

grue

nt.

The

opp

osite

ang

les

of a

!

P!

!R

and

!S

!!

Qpa

ralle

logr

am a

re c

ongr

uent

.

The

con

secu

tive

angl

es o

f a

!P

and

!S

are

supp

lem

enta

ry; !

San

d !

Rar

e su

pple

men

tary

; pa

ralle

logr

am a

re s

uppl

emen

tary

.!

Ran

d !

Qar

e su

pple

men

tary

; !Q

and

!P

are

supp

lem

enta

ry.

If a

para

llelo

gram

has

one

rig

ht

If m

!P

"90

, the

n m

!Q

"90

, m!

R"

90, a

nd m

!S

"90

.an

gle,

then

it h

as fo

ur r

ight

ang

les.

If A

BC

Dis

a p

aral

lelo

gram

,fin

d a

and

b.

A "B"

and

C"D"

are

oppo

site

sid

es,s

o A"

B"!

C"D"

.2a

"34

a"

17

!A

and

!C

are

oppo

site

ang

les,

so !

A!

!C

.8b

"11

2b

"14

Fin

d x

and

yin

eac

h p

aral

lelo

gram

.

1.2.

x%

30;y

%22

.5x

%15

;y%

11

3.4.

x%

2;y

%4

x%

10;y

%40

5.6.

x%

13;y

%32

.5x

%5;

y%

180

72x

30x

150

2y60

"55"

5x"

2y"

6x"

12x"

3y"

3y 12

6x

8y 88

6x"

4y"

3x"

BA

DC

34

2a8b

"

112"

PQ

RS

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill42

4G

lenc

oe G

eom

etry

Dia

go

nal

s o

f Pa

ralle

log

ram

sT

wo

impo

rtan

t pr

oper

ties

of

para

llelo

gram

s de

al w

ith

thei

r di

agon

als.

If A

BC

Dis

a p

aral

lelo

gram

,the

n:

The

dia

gona

ls o

f a p

aral

lelo

gram

bis

ect e

ach

othe

r.A

P"

PC

and

DP

"P

B

Eac

h di

agon

al s

epar

ates

a p

aral

lelo

gram

"

AC

D!

"C

AB

and

"A

DB

!"

CB

Din

to tw

o co

ngru

ent t

riang

les.

Fin

d x

and

yin

par

alle

logr

am A

BC

D.

The

dia

gona

ls b

isec

t ea

ch o

ther

,so

AE

"C

Ean

d D

E"

BE

.6x

"24

4y"

18x

"4

y"

4.5

Fin

d x

and

yin

eac

h p

aral

lelo

gram

.

1.2.

3.

x%

4;y

%2

x%

7;y

%14

x%

15;y

%7.

5

4.5.

6.

x%

3$1 3$ ;

y%

10!

3"x

%15

;y%

6!2"

x%

15;y

%!

241

"

Com

ple

te e

ach

sta

tem

ent

abou

t "

AB

CD

.Ju

stif

y yo

ur

answ

er.

7.!

BA

C!

!A

CD

If lin

es a

re || ,

then

alt.

int.

#ar

e #

.

8.D"

E"!

B"E"

The

diag

s.of

a p

aral

lelo

gram

bis

ect

each

oth

er.

9."

AD

C!

$C

BA

The

diag

onal

of

a pa

ralle

logr

am d

ivid

es t

he p

aral

lelo

gram

into

2 #

$s.

10.A"

D"||

C"B"

Opp

osite

sid

es o

f a

para

llelo

gram

are

|| .

AB

CD

E

x

17

4y

3x"

2y12

3x

30"

10y

2x"60

"4y

"

4x

2y28

3x4y

812

AB

E

CD

6x

1824

4y

AB

CD

P

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Para

llelo

gram

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-2

8-2

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Para

llelo

gram

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-2

8-2

©G

lenc

oe/M

cGra

w-H

ill42

5G

lenc

oe G

eom

etry

Lesson 8-2

Com

ple

te e

ach

sta

tem

ent

abou

t "

DE

FG

.Ju

stif

y yo

ur

answ

er.

1.D"

G"||

E"F";o

pp.s

ides

of

"ar

e || .

2.D"

E"!

G"F";

opp.

side

s of

"ar

e #

.

3.G"

H"!

E"H";d

iag.

of "

bise

ct e

ach

othe

r.

4.!

DE

F!

!FG

D;o

pp.#

of "

are

#.

5.!

EF

Gis

sup

plem

enta

ry t

o .

!D

EF

or !

FGD

;con

s.#

in "

are

supp

l.

6."

DG

E!

$FE

G;d

iag.

of "

sepa

rate

s "

into

2 #

$s.

ALG

EBR

AU

se "

WX

YZ

to f

ind

eac

h m

easu

re o

r va

lue.

7.m

!X

YZ

"8.

m!

WZ

Y"

9.m

!W

XY

"10

.a"

CO

OR

DIN

ATE

GEO

MET

RYF

ind

th

e co

ord

inat

es o

f th

e in

ters

ecti

on o

f th

ed

iago

nal

s of

par

alle

logr

am H

JK

Lgi

ven

eac

h s

et o

f ve

rtic

es.

11.H

(1,1

),J(

2,3)

,K(6

,3),

L(5

,1)

12.H

(!1,

4),J

(3,3

),K

(3,!

2),L

(!1,

!1)

(3.5

,2)

(1,1

)

13.P

RO

OF

Wri

te a

par

agra

ph p

roof

of

the

theo

rem

Con

secu

tive

ang

les

in a

par

alle

logr

amar

e su

pple

men

tary

.

Giv

en:"

AB

CD

Pro

ve:!

Aan

d !

Bar

e su

pple

men

tary

.!

Ban

d !

Car

e su

pple

men

tary

.!

Can

d !

Dar

e su

pple

men

tary

.!

Dan

d !

Aar

e su

pple

men

tary

.P

roof

:We

are

give

n "

AB

CD

,so

we

know

tha

t A"

B"|| C"

D"an

d B"

C"|| D"

A"by

the

defin

ition

of a

par

alle

logr

am.W

e al

so k

now

that

if tw

o pa

ralle

l lin

es a

recu

t by

a t

rans

vers

al,t

hen

cons

ecut

ive

inte

rior

ang

les

are

supp

lem

enta

ry.

So,

!A

and

!B

,!B

and

!C

,!C

and

!D

,and

!D

and

!A

are

pair

s of

supp

lem

enta

ry a

ngle

s.

DC

BA

1560

6012

0

YZ

XW

A 50"

30

2 a 70"

?

?

?

???F

G

ED

H

©G

lenc

oe/M

cGra

w-H

ill42

6G

lenc

oe G

eom

etry

Com

ple

te e

ach

sta

tem

ent

abou

t "

LM

NP

.Ju

stif

y yo

ur

answ

er.

1.L"Q"

!N"

Q";d

iag.

of "

bise

ct e

ach

othe

r.

2.!

LM

N!

!N

PL;

opp.

#of

"ar

e #

.

3."

LM

P!

$N

PM

;dia

g.of

"se

para

tes

"in

to 2

#$

s.

4.!

NP

Lis

sup

plem

enta

ry t

o .

!M

NP

or !

PLM

;con

s.#

in "

are

supp

l.

5.L"M"

!N"

P";o

pp.s

ides

of

"ar

e #

.

ALG

EBR

AU

se "

RS

TU

to f

ind

eac

h m

easu

re o

r va

lue.

6.m

!R

ST

"7.

m!

ST

U"

8.m

!T

UR

"9.

b"

CO

OR

DIN

ATE

GEO

MET

RYF

ind

th

e co

ord

inat

es o

f th

e in

ters

ecti

on o

f th

ed

iago

nal

s of

par

alle

logr

am P

RY

Zgi

ven

eac

h s

et o

f ve

rtic

es.

10.P

(2,5

),R

(3,3

),Y

(!2,

!3)

,Z(!

3,!

1)11

.P(2

,3),

R(1

,!2)

,Y(!

5,!

7),Z

(!4,

!2)

(0,1

)(!

1.5,

!2)

12.P

RO

OF

Wri

te a

par

agra

ph p

roof

of

the

follo

win

g.G

iven

:#P

RS

Tan

d #

PQ

VU

Pro

ve:!

V!

!S

Pro

of:W

e ar

e gi

ven

"P

RS

Tan

d "

PQ

VU

.Sin

ce o

ppos

itean

gles

of

a pa

ralle

logr

am a

re c

ongr

uent

,!P

#!

Van

d !

P#

!S

.Sin

ce c

ongr

uenc

e of

ang

les

is t

rans

itive

,!

V#

!S

by t

he T

rans

itive

Pro

pert

y of

Con

grue

nce.

13.C

ON

STR

UC

TIO

NM

r.R

odri

quez

use

d th

e pa

ralle

logr

am a

t th

e ri

ght

to

desi

gn a

her

ring

bone

pat

tern

for

a p

avin

g st

one.

He

will

use

the

pav

ing

ston

e fo

r a

side

wal

k.If

m!

1 is

130

,fin

d m

!2,

m!

3,an

d m

!4.

50,1

30,5

0

12

34P

R

S

UVQ

T

612

5

5512

5T

U

SR

B30

"23

4b !

1

25"

?

?

??

?P

N

MQ

L

Pra

ctic

e (A

vera

ge)

Para

llelo

gram

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-2

8-2



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csPa

ralle

logr

ams

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-2

8-2

©G

lenc

oe/M

cGra

w-H

ill42

7G

lenc

oe G

eom

etry

Lesson 8-2

Pre-

Act

ivit

yH

ow a

re p

aral

lelo

gram

s u

sed

to

rep

rese

nt

dat

a?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

2 at

the

top

of

page

411

in y

our

text

book

.

•W

hat

is t

he n

ame

of t

he s

hape

of

the

top

surf

ace

of e

ach

wed

ge o

f ch

eese

?pa

ralle

logr

am•

Are

the

thr

ee p

olyg

ons

show

n in

the

dra

win

g si

mila

r po

lygo

ns?

Exp

lain

you

rre

ason

ing.

No;

sam

ple

answ

er:T

heir

sid

es a

re n

ot p

ropo

rtio

nal.

Rea

din

g t

he

Less

on

1.U

nder

line

wor

ds o

r ph

rase

s th

at c

an c

ompl

ete

the

follo

win

g se

nten

ces

to m

ake

stat

emen

tsth

at a

re a

lway

s tr

ue.(

The

re m

ay b

e m

ore

than

one

cor

rect

cho

ice

for

som

e of

the

sen

tenc

es.)

a.O

ppos

ite

side

s of

a p

aral

lelo

gram

are

(co

ngru

ent/

perp

endi

cula

r/pa

ralle

l).

b.C

onse

cuti

ve a

ngle

s of

a p

aral

lelo

gram

are

(co

mpl

emen

tary

/sup

plem

enta

ry/c

ongr

uent

).

c.A

dia

gona

l of

a pa

ralle

logr

am d

ivid

es t

he p

aral

lelo

gram

into

tw

o(a

cute

/rig

ht/o

btus

e/co

ngru

ent)

tri

angl

es.

d.O

ppos

ite

angl

es o

f a

para

llelo

gram

are

(co

mpl

emen

tary

/sup

plem

enta

ry/c

ongr

uent

).

e.T

he d

iago

nals

of

a pa

ralle

logr

am (

bise

ct e

ach

othe

r/ar

e pe

rpen

dicu

lar/

are

cong

ruen

t).

f.If

a p

aral

lelo

gram

has

one

rig

ht a

ngle

,the

n al

l of

its

othe

r an

gles

are

(acu

te/r

ight

/obt

use)

ang

les.

2.L

et A

BC

Dbe

a p

aral

lelo

gram

wit

h A

B$

BC

and

wit

h no

rig

ht a

ngle

s.

a.Sk

etch

a p

aral

lelo

gram

tha

t m

atch

es t

he d

escr

ipti

onS

ampl

e an

swer

:ab

ove

and

draw

dia

gona

l B"D"

.

In p

arts

b–f

,com

plet

e ea

ch s

ente

nce.

b.A"

B"||

and

A"D"

||.

c.A"

B"!

and

B "C"

!B

.

d.!

A!

and

!A

BC

!.

e.!

AD

B!

beca

use

thes

e tw

o an

gles

are

angl

es f

orm

ed b

y th

e tw

o pa

ralle

l lin

es

and

and

the

tran

sver

sal

.

f."

AB

D!

.

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

new

the

orem

s in

geo

met

ry is

to

rela

te t

hem

to

theo

rem

s yo

ule

arne

d ea

rlie

r.N

ame

a th

eore

m a

bout

par

alle

l lin

es t

hat

can

be u

sed

to r

emem

ber

the

theo

rem

tha

t sa

ys,“

If a

par

alle

logr

am h

as o

ne r

ight

ang

le,i

t ha

s fo

ur r

ight

ang

les.

”P

erpe

ndic

ular

Tra

nsve

rsal

The

orem

$C

DBBD

!"#

BC

!"#

AD

!"#

inte

rior

alte

rnat

e!

CB

D

!C

DA

!C

A"D"

C"D"

B"C"

C"D"

DA

CB

©G

lenc

oe/M

cGra

w-H

ill42

8G

lenc

oe G

eom

etry

Tess

ella

tions

A t

esse

llat

ion

is a

tili

ng p

atte

rn m

ade

of p

olyg

ons.

The

pat

tern

can

be

ext

ende

d so

tha

t th

e po

lygo

nal t

iles

cove

r th

e pl

ane

com

plet

ely

wit

hno

gap

s.A

che

cker

boar

d an

d a

hone

ycom

b pa

tter

n ar

e ex

ampl

es o

fte

ssel

lati

ons.

Som

etim

es t

he s

ame

poly

gon

can

mak

e m

ore

than

one

tess

ella

tion

pat

tern

.Bot

h pa

tter

ns b

elow

can

be

form

ed f

rom

an

isos

cele

s tr

iang

le.

Dra

w a

tes

sell

atio

n u

sin

g ea

ch p

olyg

on.

1.2.

3.

Dra

w t

wo

dif

fere

nt

tess

ella

tion

s u

sin

g ea

ch p

olyg

on.

4.5.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-2

8-2



Stu

dy

Gu

ide

and I

nte

rven

tion

Test

s fo

r Pa

ralle

logr

ams

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-3

8-3

©G

lenc

oe/M

cGra

w-H

ill42

9G

lenc

oe G

eom

etry

Lesson 8-3

Co

nd

itio

ns

for

a Pa

ralle

log

ram

The

re a

re m

any

way

s to

es

tabl

ish

that

a q

uadr

ilate

ral i

s a

para

llelo

gram

.

If:If:

both

pai

rs o

f opp

osite

sid

es a

re p

aral

lel,

A"B"|| D"

C"an

d A"D"

|| B"C"

,

both

pai

rs o

f opp

osite

sid

es a

re c

ongr

uent

,A"B"

! D"

C"an

d A"D"

! B"

C",

both

pai

rs o

f opp

osite

ang

les

are

cong

ruen

t,!

AB

C!

!A

DC

and

!D

AB

!!

BC

D,

the

diag

onal

s bi

sect

eac

h ot

her,

A"E"!

C"E"

and

D"E"

! B"

E",

one

pair

of o

ppos

ite s

ides

is c

ongr

uent

and

par

alle

l,A"B"

|| C"D"

and

A"B"!

C"D"

, or

A"D"|| B"

C"an

d A"D"

!B"C"

,

then

:th

e fig

ure

is a

par

alle

logr

am.

then

:A

BC

Dis

a p

aral

lelo

gram

.

Fin

d x

and

yso

th

at F

GH

Jis

a

par

alle

logr

am.

FG

HJ

is a

par

alle

logr

am if

the

leng

ths

of t

he o

ppos

ite

side

s ar

e eq

ual.

6x#

3 "

154x

!2y

"2

6x"

124(

2) !

2y"

2x

"2

8 !

2y"

2!

2y"

!6

y"

3

Fin

d x

and

yso

th

at e

ach

qu

adri

late

ral

is a

par

alle

logr

am.

1.x

%7;

y%

42.

x%

5;y

%25

3.x

%31

;y%

54.

x%

5;y

%3

5.x

%15

;y%

96.

x%

30;y

%15

6y"

3 x"

( x #

y) "

2x"

30"

24"

9x"

6y45

"18

5x"

5y"

25"

11x"

5y"

55"

2x !

2

2 y8

12

JHG

F6x

# 3

154x

! 2

y2

AB

CD

E

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill43

0G

lenc

oe G

eom

etry

Para

llelo

gra

ms

on

th

e C

oo

rdin

ate

Plan

eO

n th

e co

ordi

nate

pla

ne,t

he D

ista

nce

Form

ula

and

the

Slop

e Fo

rmul

a ca

n be

use

d to

tes

t if

a q

uadr

ilate

ral i

s a

para

llelo

gram

.

Det

erm

ine

wh

eth

er A

BC

Dis

a p

aral

lelo

gram

.T

he v

erti

ces

are

A(!

2,3)

,B(3

,2),

C(2

,!1)

,and

D(!

3,0)

.

Met

hod

1:U

se t

he S

lope

For

mul

a,m

"%y x2 2

! !

y x1 1%

.

slop

e of

A "D"

"% !

23 !!(0 !

3)%

"%3 1%

"3

slop

e of

B"C"

"%2

3!!(!

21)%

"%3 1%

"3

slop

e of

A "B"

"% 3

2 !! (!

3 2)%

"!

%1 5%sl

ope

of C"

D""

% 2!!1

(! !0 3)

%"

!%1 5%

Opp

osit

e si

des

have

the

sam

e sl

ope,

so A"

B"|| C"

D"an

d A"

D"|| B"

C".B

oth

pair

s of

opp

osit

e si

des

are

para

llel,

so A

BC

Dis

a p

aral

lelo

gram

.

Met

hod

2:U

se t

he D

ista

nce

Form

ula,

d"

#(x

2!

"x 1

)2#

"(y

2!

"y 1

)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"

Bot

h pa

irs

of o

ppos

ite

side

s ha

ve t

he s

ame

leng

th,s

o A

BC

Dis

a p

aral

lelo

gram

.

Det

erm

ine

wh

eth

er a

fig

ure

wit

h t

he

give

n v

erti

ces

is a

par

alle

logr

am.U

se 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);

Slop

e Fo

rmul

aSl

ope

Form

ula

yes

yes

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

Form

ula

Dis

tanc

e an

d Sl

ope

Form

ulas

noye

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

tanc

e an

d Sl

ope

Form

ulas

Mid

poin

t Fo

rmul

a

yes

no

7.A

par

alle

logr

am h

as v

erti

ces

R(!

2,!

1),S

(2,1

),an

d T

(0,!

3).F

ind

all p

ossi

ble

coor

dina

tes

for

the

four

th v

erte

x.

(4,!

1),(

0,3)

,or

(!4,

!5)

x

y

O

C

A

D

B

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Test

s fo

r Pa

ralle

logr

ams

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-3

8-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Test

s fo

r Pa

ralle

logr

ams

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-3

8-3

©G

lenc

oe/M

cGra

w-H

ill43

1G

lenc

oe G

eom

etry

Lesson 8-3

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.

Yes;

a pa

ir o

f op

posi

te s

ides

Yes;

both

pai

rs o

f op

posi

teis

par

alle

l and

con

grue

nt.

angl

es a

re c

ongr

uent

.

3.4.

No;

none

of

the

test

s fo

r Ye

s;bo

th p

airs

of

oppo

site

sid

espa

ralle

logr

ams

is f

ulfil

led.

are

cong

ruen

t.

CO

OR

DIN

ATE

GEO

MET

RYD

eter

min

e w

het

her

a f

igu

re w

ith

th

e gi

ven

ver

tice

s is

ap

aral

lelo

gram

.Use

th

e m

eth

od i

nd

icat

ed.

5.P

(0,0

),Q

(3,4

),S

(7,4

),Y

(4,0

);Sl

ope

Form

ula

yes

6.S

(!2,

1),R

(1,3

),T

(2,0

),Z

(!1,

!2)

;Dis

tanc

e an

d Sl

ope

Form

ula

yes

7.W

(2,5

),R

(3,3

),Y

(!2,

!3)

,N(!

3,1)

;Mid

poin

t Fo

rmul

ano

ALG

EBR

AF

ind

xan

d y

so t

hat

eac

h q

uad

rila

tera

l is

a p

aral

lelo

gram

.

8.9.

x%

24,y

%19

x%

3,y

%14

10.

11.

x%

45,y

%20

x%

17,y

%9

3x !

14

x #

20

3y #

2y

# 2

0( 4

x !

35)

"

( 3x

# 1

0)"

( 2y

! 5

) "

( y #

15)

"

2y !

11

y # 3

4x ! 3

3x

x #

16

y #

19

2y

2x !

8

©G

lenc

oe/M

cGra

w-H

ill43

2G

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.

Yes;

the

diag

onal

s bi

sect

No;

none

of

the

test

s fo

rea

ch o

ther

.pa

ralle

logr

ams

is f

ulfil

led.

3.4.

Yes;

both

pai

rs o

f op

posi

teN

o;no

ne o

f th

e te

sts

for

angl

es a

re c

ongr

uent

.pa

ralle

logr

ams

is f

ulfil

led.

CO

OR

DIN

ATE

GEO

MET

RYD

eter

min

e w

het

her

a f

igu

re w

ith

th

e gi

ven

ver

tice

s is

ap

aral

lelo

gram

.Use

th

e m

eth

od i

nd

icat

ed.

5.P

(!5,

1),S

(!2,

2),F

(!1,

!3)

,T(2

,!2)

;Slo

pe F

orm

ula

no

6.R

(!2,

5),O

(1,3

),M

(!3,

!4)

,Y(!

6,!

2);D

ista

nce

and

Slop

e Fo

rmul

aye

s

ALG

EBR

AF

ind

xan

d y

so t

hat

eac

h q

uad

rila

tera

l is

a p

aral

lelo

gram

.

7.8.

x%

20,y

%12

x%

!6,

y%

13

9.10

.

x%

!3,

y%

2x

%!

2,y

%!

5

11.T

ILE

DES

IGN

The

pat

tern

sho

wn

in t

he f

igur

e is

to

cons

ist

of c

ongr

uent

pa

ralle

logr

ams.

How

can

the

des

igne

r be

cer

tain

tha

t th

e sh

apes

are

para

llelo

gram

s?S

ampl

e an

swer

:Con

firm

tha

t bo

th p

airs

of

oppo

site

!s

are

#.

!4y

! 2

x # 12

!2x

# 6

y # 2

3!

4x #

6

!6x

7y #

312

y !

7

3y !

5!

3x

# 4

!4x

! 2

2y #

8( 5

x #

29)

"

( 7x

! 1

1)"

( 3y

# 1

5)"

( 5y

! 9

) "

118"

62"

118"

62"

Pra

ctic

e (A

vera

ge)

Test

s fo

r Pa

ralle

logr

ams

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-3

8-3



Rea

din

g t

o L

earn

Math

emati

csTe

sts

for

Para

llelo

gram

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-3

8-3

©G

lenc

oe/M

cGra

w-H

ill43

3G

lenc

oe G

eom

etry

Lesson 8-3

Pre-

Act

ivit

yH

ow a

re p

aral

lelo

gram

s u

sed

in

arc

hit

ectu

re?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

3 at

the

top

of

page

417

in y

our

text

book

.

Mak

e tw

o ob

serv

atio

ns a

bout

the

ang

les

in t

he r

oof

of t

he c

over

ed b

ridg

e.S

ampl

e an

swer

:Opp

osite

ang

les

appe

ar c

ongr

uent

.C

onse

cutiv

e an

gles

app

ear

supp

lem

enta

ry.

Rea

din

g t

he

Less

on

1.W

hich

of

the

follo

win

g co

ndit

ions

gua

rant

ee t

hat

a qu

adri

late

ral i

s a

para

llelo

gram

?

A.T

wo

side

s ar

e pa

ralle

l.B

,D,G

,H,J

,LB

.Bot

h pa

irs

of o

ppos

ite

side

s ar

e co

ngru

ent.

C.T

he d

iago

nals

are

per

pend

icul

ar.

D.A

pai

r of

opp

osit

e si

des

is b

oth

para

llel a

nd c

ongr

uent

.E

.The

re a

re t

wo

righ

t an

gles

.F.

The

sum

of

the

mea

sure

s of

the

inte

rior

ang

les

is 3

60.

G.A

ll fo

ur s

ides

are

con

grue

nt.

H.B

oth

pair

s of

opp

osit

e an

gles

are

con

grue

nt.

I.T

wo

angl

es a

re a

cute

and

the

oth

er t

wo

angl

es a

re o

btus

e.J.

The

dia

gona

ls b

isec

t ea

ch o

ther

.K

.The

dia

gona

ls a

re c

ongr

uent

.L

.A

ll fo

ur a

ngle

s ar

e ri

ght

angl

es.

2.D

eter

min

e w

heth

er t

here

is e

noug

h gi

ven

info

rmat

ion

to k

now

tha

t ea

ch f

igur

e is

apa

ralle

logr

am.I

f so

,sta

te t

he d

efin

itio

n or

the

orem

tha

t ju

stif

ies

your

con

clus

ion.

a.b.

noYe

s;if

the

diag

onal

s bi

sect

eac

hot

her,

then

the

qua

drila

tera

l is

apa

ralle

logr

am.

c.d.

Yes;

if bo

th p

airs

of

oppo

site

no

side

s ar

e #

,the

n qu

adri

late

ral

is "

.

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

larg

e nu

mbe

r of

mat

hem

atic

al id

eas

is t

o th

ink

of t

hem

ingr

oups

.How

can

you

sta

te t

he c

ondi

tion

s as

one

gro

up a

bout

the

sid

esof

qua

drila

tera

lsth

at g

uara

ntee

tha

t th

e qu

adri

late

ral i

s a

para

llelo

gram

?S

ampl

e an

swer

:bot

hpa

irs

of o

ppos

ite s

ides

par

alle

l,bo

th p

airs

of

oppo

site

sid

es c

ongr

uent

,or

one

pai

r of

opp

osite

sid

es b

oth

para

llel a

nd c

ongr

uent

©G

lenc

oe/M

cGra

w-H

ill43

4G

lenc

oe G

eom

etry

Test

s fo

r Pa

ralle

logr

ams

By

defi

niti

on,a

qua

drila

tera

l is

a pa

ralle

logr

am if

and

onl

y if

both

pai

rs o

f op

posi

te s

ides

are

para

llel.

Wha

t co

ndit

ions

oth

er t

han

both

pai

rs o

f op

posi

te s

ides

par

alle

l will

gua

rant

eeth

at a

qua

drila

tera

l is

a pa

ralle

logr

am?

In t

his

acti

vity

,sev

eral

pos

sibi

litie

s w

ill b

ein

vest

igat

ed b

y dr

awin

g qu

adri

late

rals

to

sati

sfy

cert

ain

cond

itio

ns.R

emem

ber

that

any

te

st t

hat

seem

s to

wor

k is

not

gua

rant

eed

to w

ork

unle

ss it

can

be

form

ally

pro

ven.

Com

ple

te.

1.D

raw

a q

uadr

ilate

ral w

ith

one

pair

of

oppo

site

sid

es c

ongr

uent

.M

ust

it b

e a

para

llelo

gram

?no

2.D

raw

a q

uadr

ilate

ral w

ith

both

pai

rs o

f op

posi

te s

ides

con

grue

nt.

Mus

t it

be

a pa

ralle

logr

am?

yes

3.D

raw

a q

uadr

ilate

ral w

ith

one

pair

of

oppo

site

sid

es p

aral

lel a

nd

the

othe

r pa

ir o

f op

posi

te s

ides

con

grue

nt.M

ust

it b

e a

para

llelo

gram

?no

4.D

raw

a q

uadr

ilate

ral w

ith

one

pair

of

oppo

site

sid

es b

oth

para

llel

and

cong

ruen

t.M

ust

it b

e a

para

llelo

gram

?ye

s

5.D

raw

a q

uadr

ilate

ral w

ith

one

pair

of

oppo

site

ang

les

cong

ruen

t.M

ust

it b

e a

para

llelo

gram

?no

6.D

raw

a q

uadr

ilate

ral w

ith

both

pai

rs o

f op

posi

te a

ngle

s co

ngru

ent.

Mus

t it

be

a pa

ralle

logr

am?

yes

7.D

raw

a q

uadr

ilate

ral w

ith

one

pair

of

oppo

site

sid

es p

aral

lel a

nd

one

pair

of

oppo

site

ang

les

cong

ruen

t.M

ust

it b

e a

para

llelo

gram

?ye

s

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-3

8-3



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Rec

tang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-4

8-4

©G

lenc

oe/M

cGra

w-H

ill43

5G

lenc

oe G

eom

etry

Lesson 8-4

Pro

per

ties

of

Rec

tan

gle

sA

rec

tan

gle

is a

qua

drila

tera

l wit

h fo

ur

righ

t an

gles

.Her

e ar

e th

e pr

oper

ties

of

rect

angl

es.

A r

ecta

ngle

has

all

the

prop

erti

es o

f a

para

llelo

gram

.

•O

ppos

ite

side

s ar

e pa

ralle

l.•

Opp

osit

e an

gles

are

con

grue

nt.

•O

ppos

ite

side

s ar

e co

ngru

ent.

•C

onse

cuti

ve a

ngle

s ar

e su

pple

men

tary

.•

The

dia

gona

ls b

isec

t ea

ch o

ther

.

Als

o:

•A

ll fo

ur a

ngle

s ar

e ri

ght

angl

es.

!U

TS

,!T

SR

,!S

RU

,and

!R

UT

are

righ

t an

gles

.•

The

dia

gona

ls a

re c

ongr

uent

.T "

R"!

U"S"

TS R

U

Q

In r

ecta

ngl

e R

ST

Uab

ove,

US

%6x

#3

and

RT

%7x

!2.

Fin

d x

.T

he d

iago

nals

of

a re

ctan

gle

bise

ct e

ach

othe

r,so

US

"R

T.

6x#

3"

7x!

23

"x

!2

5"

x

In r

ecta

ngl

e R

ST

Uab

ove,

m!

ST

R%

8x#

3 an

d m

!U

TR

%16

x!

9.F

ind

m!

ST

R.

!U

TS

is a

rig

ht a

ngle

,so

m!

ST

R#

m!

UT

R"

90.

8x#

3 #

16x

!9

"90

24x

!6

"90

24x

"96

x"

4

m!

ST

R"

8x#

3 "

8(4)

#3

or 3

5

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

AB

CD

is a

rec

tan

gle.

1.If

AE

"36

and

CE

"2x

!4,

find

x.

20

2.If

BE

"6y

#2

and

CE

"4y

#6,

find

y.

2

3.If

BC

"24

and

AD

"5y

!1,

find

y.

5

4.If

m!

BE

A"

62,f

ind

m!

BA

C.

59

5.If

m!

AE

D"

12x

and

m!

BE

C"

10x

#20

,fin

d m

!A

ED

.12

0

6.If

BD

"8y

!4

and

AC

"7y

#3,

find

BD

.52

7.If

m!

DB

C"

10x

and

m!

AC

B"

4x2 !

6,fi

nd m

!A

CB

.30

8.If

AB

"6y

and

BC

"8y

,fin

d B

Din

ter

ms

of y

.10

y

9.In

rec

tang

le M

NO

P,m

!1

"40

.Fin

d th

e m

easu

re o

f ea

ch

num

bere

d an

gle.

m!

2 %

40;m

!3

%50

;m!

4 %

50;m

!5

%80

;m!

6 %

100

MN O

P

K 12

345

6

BC D

A

E

©G

lenc

oe/M

cGra

w-H

ill43

6G

lenc

oe G

eom

etry

Pro

ve t

hat

Par

alle

log

ram

s A

re R

ecta

ng

les

The

dia

gona

ls o

f a

rect

angl

e ar

eco

ngru

ent,

and

the

conv

erse

is a

lso

true

.

If th

e di

agon

als

of a

par

alle

logr

am a

re c

ongr

uent

, the

n th

e pa

ralle

logr

am is

a r

ecta

ngle

.

In t

he c

oord

inat

e pl

ane

you

can

use

the

Dis

tanc

e Fo

rmul

a,th

e Sl

ope

Form

ula,

and

prop

erti

es o

f di

agon

als

to s

how

tha

t a

figu

re is

a r

ecta

ngle

.

Det

erm

ine

wh

eth

er A

(!3,

0),B

(!2,

3),C

(4,1

),an

d D

(3,!

2) a

re t

he

vert

ices

of

a re

ctan

gle.

Met

hod

1:U

se t

he S

lope

For

mul

a.

slop

e of

A "B"

"% !

23 !!(0 !

3)%

"%3 1%

or 3

slop

e of

A"D"

"% 3!

!2(! !

0 3)%

"%! 62 %

or !

%1 3%

slop

e of

C"D"

"%! 32 !!

41%

"%! !

3 1%or

3sl

ope

of B"

C""

% 41 !

! (!3 2)

%"

%! 62 %or

!%1 3%

Opp

osit

e si

des

are

para

llel,

so t

he f

igur

e is

a p

aral

lelo

gram

.Con

secu

tive

sid

es a

repe

rpen

dicu

lar,

so A

BC

Dis

a r

ecta

ngle

.

Met

hod

2:U

se t

he M

idpo

int

and

Dis

tanc

e Fo

rmul

as.

The

mid

poin

t of

A"C"

is $%!

3 2#4

%,%

0# 2

1%

%"$%1 2% ,

%1 2% %and

the

mid

poin

t of

B"D"

is

$%!2 2#

3%

,%3

! 22

%%"

$%1 2% ,%1 2% %.

The

dia

gona

ls h

ave

the

sam

e m

idpo

int

so t

hey

bise

ct e

ach

othe

r.

Thu

s,A

BC

Dis

a p

aral

lelo

gram

.

AC

"#

(!3

!"

4)2

#"

(0 !

"1)

2"

"#

49 #

"1"

or #

50"B

D"

#(!

2 !

"3)

2#

"(3

!"

(!2)

)2"

"#

25 #

"25"

or #

50"T

he d

iago

nals

are

con

grue

nt.A

BC

Dis

a p

aral

lelo

gram

wit

h di

agon

als

that

bis

ect

each

othe

r,so

it is

a r

ecta

ngle

.

Det

erm

ine

wh

eth

er A

BC

Dis

a r

ecta

ngl

e gi

ven

eac

h s

et o

f ve

rtic

es.J

ust

ify

you

ran

swer

.S

ampl

e ju

stifi

catio

ns a

re g

iven

.

1.A

(!3,

1),B

(!3,

3),C

(3,3

),D

(3,1

)2.

A(!

3,0)

,B(!

2,3)

,C(4

,5),

D(3

,2)

Yes;

cons

ecut

ive

side

s ar

e ⊥.

No;

cons

ecut

ive

side

s ar

e no

t ⊥.

3.A

(!3,

0),B

(!2,

2),C

(3,0

),D

(2,!

2)4.

A(!

1,0)

,B(0

,2),

C(4

,0),

D(3

,!2)

No;

cons

ecut

ive

side

s ar

e no

t ⊥.

Yes;

cons

ecut

ive

side

s ar

e ⊥.

5.A

(!1,

!5)

,B(!

3,0)

,C(2

,2),

D(4

,!3)

6.A

(!1,

!1)

,B(0

,2),

C(4

,3),

D(3

,0)

Yes;

the

diag

onal

s ar

e co

ngru

ent

No;

the

diag

onal

s ar

e no

t #

.an

d bi

sect

eac

h ot

her.

7.A

par

alle

logr

am h

as v

erti

ces

R(!

3,!

1),S

(!1,

2),a

nd T

(5,!

2).F

ind

the

coor

dina

tes

ofU

so t

hat

RS

TU

is a

rec

tang

le.

(3,!

5)

x

y O

D

B

A

EC

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Rec

tang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-4

8-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Rec

tang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-4

8-4

©G

lenc

oe/M

cGra

w-H

ill43

7G

lenc

oe G

eom

etry

Lesson 8-4

ALG

EBR

AA

BC

Dis

a r

ecta

ngl

e.

1.If

AC

"2x

#13

and

DB

"4x

!1,

find

x.

7

2.If

AC

"x

#3

and

DB

"3x

!19

,fin

d A

C.

14

3.If

AE

"3x

#3

and

EC

"5x

!15

,fin

d A

C.

60

4.If

DE

"6x

!7

and

AE

"4x

#9,

find

DB

.82

5.If

m!

DA

C"

2x#

4 an

d m

!B

AC

"3x

#1,

find

x.

17

6.If

m!

BD

C"

7x#

1 an

d m

!A

DB

"9x

!7,

find

m!

BD

C.

43

7.If

m!

AB

D"

x2!

7 an

d m

!C

DB

"4x

#5,

find

x.

6

8.If

m!

BA

C"

x2#

3 an

d m

!C

AD

"x

#15

,fin

d m

!B

AC

.67

or

84

PR

ST

is a

rec

tan

gle.

Fin

d e

ach

mea

sure

if

m!

1 %

50.

9.m

!2

4010

.m!

340

11.m

!4

5012

.m!

540

13.m

!6

4014

.m!

750

15.m

!8

100

16.m

!9

80

CO

OR

DIN

ATE

GEO

MET

RYD

eter

min

e w

het

her

TU

XY

is a

rec

tan

gle

give

n e

ach

set

of v

erti

ces.

Just

ify

you

r an

swer

.

17.T

(!3,

!2)

,U(!

4,2)

,X(2

,4),

Y(3

,0)

No;

sam

ple

answ

er:A

ngle

s ar

e no

t ri

ght

angl

es.

18.T

(!6,

3),U

(0,6

),X

(2,2

),Y

(!4,

!1)

Yes;

sam

ple

answ

er:O

ppos

ite s

ides

are

con

grue

nt a

nd d

iago

nals

are

cong

ruen

t.

19.T

(4,1

),U

(3,!

1),X

(!3,

2),Y

(!2,

4)Ye

s;sa

mpl

e an

swer

:Opp

osite

sid

es a

re p

aral

lel a

nd c

onse

cutiv

e si

des

are

perp

endi

cula

r.

T

12

34

56

78

9

SRPD

CBA

E

©G

lenc

oe/M

cGra

w-H

ill43

8G

lenc

oe G

eom

etry

ALG

EBR

AR

ST

Uis

a r

ecta

ngl

e.

1.If

UZ

"x

#21

and

ZS

"3x

!15

,fin

d U

S.

78

2.If

RZ

"3x

#8

and

ZS

"6x

!28

,fin

d U

Z.

44

3.If

RT

"5x

#8

and

RZ

"4x

#1,

find

ZT

.9

4.If

m!

SU

T"

3x#

6 an

d m

!R

US

"5x

!4,

find

m!

SU

T.

39

5.If

m!

SR

T"

x2#

9 an

d m

!U

TR

"2x

#44

,fin

d x.

!5

or 7

6.If

m!

RS

U"

x2!

1 an

d m

!T

US

"3x

#9,

find

m!

RS

U.

24 o

r 3

GH

JK

is a

rec

tan

gle.

Fin

d e

ach

mea

sure

if

m!

1 %

37.

7.m

!2

538.

m!

337

9.m

!4

3710

.m!

553

11.m

!6

106

12.m

!7

74

CO

OR

DIN

ATE

GEO

MET

RYD

eter

min

e w

het

her

BG

HL

is a

rec

tan

gle

give

n e

ach

set

of v

erti

ces.

Just

ify

you

r an

swer

.

13.B

(!4,

3),G

(!2,

4),H

(1,!

2),L

(!1,

!3)

Yes;

sam

ple

answ

er:O

ppos

ite s

ides

are

par

alle

l and

con

secu

tive

side

sar

e pe

rpen

dicu

lar.

14.B

(!4,

5),G

(6,0

),H

(3,!

6),L

(!7,

!1)

Yes;

sam

ple

answ

er:O

ppos

ite s

ides

are

con

grue

nt a

nd d

iago

nals

are

cong

ruen

t.

15.B

(0,5

),G

(4,7

),H

(5,4

),L

(1,2

)N

o;sa

mpl

e an

swer

:Dia

gona

ls a

re n

ot c

ongr

uent

.

16.L

AN

DSC

API

NG

Hun

ting

ton

Park

off

icia

ls a

ppro

ved

a re

ctan

gula

r pl

ot o

f la

nd f

or a

Japa

nese

Zen

gar

den.

Is it

suf

fici

ent

to k

now

tha

t op

posi

te s

ides

of

the

gard

en p

lot

are

cong

ruen

t an

d pa

ralle

l to

dete

rmin

e th

at t

he g

arde

n pl

ot is

rec

tang

ular

? E

xpla

in.

No;

if yo

u on

ly k

now

tha

t op

posi

te s

ides

are

con

grue

nt a

nd p

aral

lel,

the

mos

t yo

u ca

n co

nclu

de is

tha

t th

e pl

ot is

a p

aral

lelo

gram

.

K

21

5

43

67

JHGU

TSR

Z

Pra

ctic

e (A

vera

ge)

Rec

tang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-4

8-4



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csR

ecta

ngle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-4

8-4

©G

lenc

oe/M

cGra

w-H

ill43

9G

lenc

oe G

eom

etry

Lesson 8-4

Pre-

Act

ivit

yH

ow a

re r

ecta

ngl

es u

sed

in

ten

nis

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

4 at

the

top

of

page

424

in y

our

text

book

.

Are

the

sin

gles

cou

rt a

nd d

oubl

es c

ourt

sim

ilar

rect

angl

es?

Exp

lain

you

ran

swer

.N

o;sa

mpl

e an

swer

:All

of t

heir

ang

les

are

cong

ruen

t,bu

t th

eir

side

s ar

e no

t pr

opor

tiona

l.Th

e do

uble

s co

urt

is w

ider

in r

elat

ion

to it

s le

ngth

.

Rea

din

g t

he

Less

on

1.D

eter

min

e w

heth

er e

ach

sent

ence

is a

lway

s,so

met

imes

,or

neve

rtr

ue.

a.If

a q

uadr

ilate

ral h

as f

our

cong

ruen

t an

gles

,it

is a

rec

tang

le.

alw

ays

b.If

con

secu

tive

ang

les

of a

qua

drila

tera

l are

sup

plem

enta

ry,t

hen

the

quad

rila

tera

l is

a r

ecta

ngle

.so

met

imes

c.T

he d

iago

nals

of

a re

ctan

gle

bise

ct e

ach

othe

r.al

way

sd.

If t

he d

iago

nals

of

a qu

adri

late

ral b

isec

t ea

ch o

ther

,the

qua

drila

tera

l is

a re

ctan

gle.

som

etim

ese.

Con

secu

tive

ang

les

of a

rec

tang

le a

re c

ompl

emen

tary

.ne

ver

f.C

onse

cuti

ve a

ngle

s of

a r

ecta

ngle

are

con

grue

nt.

alw

ays

g.If

the

dia

gona

ls o

f a

quad

rila

tera

l are

con

grue

nt,t

he q

uadr

ilate

ral i

s a

rect

angl

e.so

met

imes

h.

A d

iago

nal o

f a

rect

angl

e bi

sect

s tw

o of

its

angl

es.

som

etim

esi.

A d

iago

nal o

f a

rect

angl

e di

vide

s th

e re

ctan

gle

into

tw

o co

ngru

ent

righ

t tr

iang

les.

alw

ays

j.If

the

dia

gona

ls o

f a

quad

rila

tera

l bis

ect

each

oth

er a

nd a

re c

ongr

uent

,the

quad

rila

tera

l is

a re

ctan

gle.

alw

ays

k.If

a p

aral

lelo

gram

has

one

rig

ht a

ngle

,it

is a

rec

tang

le.

alw

ays

l.If

a p

aral

lelo

gram

has

fou

r co

ngru

ent

side

s,it

is a

rec

tang

le.

som

etim

es

2.A

BC

Dis

a r

ecta

ngle

wit

h A

D&

AB

.N

ame

each

of

the

follo

win

g in

thi

s fi

gure

.

a.al

l seg

men

ts t

hat

are

cong

ruen

t to

B"E"

A"E",

C"E",

D"E"

b.al

l ang

les

cong

ruen

t to

!1

!2,

!5,

!6

c.al

l ang

les

cong

ruen

t to

!7

!3,

!4,

!8

d.tw

o pa

irs

of c

ongr

uent

tri

angl

es$

AE

D#

$B

EC

,$A

EB

#$

DE

C,

$B

CD

#$

DA

B,$

AB

C#

$C

DA

Hel

pin

g Y

ou

Rem

emb

er3.

It is

eas

ier

to r

emem

ber

a la

rge

num

ber

of g

eom

etri

c re

lati

onsh

ips

and

theo

rem

s if

you

are

able

to

com

bine

som

e of

the

m.H

ow c

an y

ou c

ombi

ne t

he t

wo

theo

rem

s ab

out

diag

onal

s th

at y

ou s

tudi

ed in

thi

s le

sson

?S

ampl

e an

swer

:A p

aral

lelo

gram

is a

rect

angl

e if

and

only

if it

s di

agon

als

are

cong

ruen

t.

A12

34

5 67

8DC

BE

©G

lenc

oe/M

cGra

w-H

ill44

0G

lenc

oe G

eom

etry

Cou

ntin

g S

quar

es a

nd R

ecta

ngle

sE

ach

puzz

le b

elow

con

tain

s m

any

squa

res

and/

or r

ecta

ngle

s.C

ount

the

m c

aref

ully

.You

may

wan

t to

labe

l eac

h re

gion

so

you

can

list

all p

ossi

bilit

ies.

How

man

y re

ctan

gles

are

in

th

e fi

gure

at

th

e ri

ght?

Lab

el e

ach

smal

l reg

ion

wit

h a

lett

er.T

hen

list

each

rec

tang

le

by w

riti

ng t

he le

tter

s of

reg

ions

it c

onta

ins.

A,B

,C,D

,AB

,CD

,AC

,BD

,AB

CD

The

re a

re 9

rec

tang

les.

How

man

y sq

uar

es a

re i

n e

ach

fig

ure

?

1.16

2.10

3.8

How

man

y re

ctan

gles

are

in

eac

h f

igu

re?

1.19

2.25

3.21

AB

CD

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-4

8-4

Exam

ple

Exam

ple



Stu

dy

Gu

ide

and I

nte

rven

tion

Rho

mbi

and

Squ

ares

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-5

8-5

©G

lenc

oe/M

cGra

w-H

ill44

1G

lenc

oe G

eom

etry

Lesson 8-5

Pro

per

ties

of

Rh

om

bi

A r

hom

bus

is a

qua

drila

tera

l wit

h fo

ur

cong

ruen

t si

des.

Opp

osit

e si

des

are

cong

ruen

t,so

a r

hom

bus

is a

lso

apa

ralle

logr

am a

nd h

as a

ll of

the

pro

pert

ies

of a

par

alle

logr

am.R

hom

bi

also

hav

e th

e fo

llow

ing

prop

erti

es.

The

dia

gona

ls a

re p

erpe

ndic

ular

.M"

H"⊥

R"O"

Eac

h di

agon

al b

isec

ts a

pai

r of

opp

osite

ang

les.

M"H"

bise

cts

!R

MO

and

!R

HO

.R"

O"bi

sect

s !

MR

Han

d !

MO

H.

If th

e di

agon

als

of a

par

alle

logr

am a

re

If R

HO

Mis

a p

aral

lelo

gram

and

R"O"

⊥M"

H",

perp

endi

cula

r, th

en th

e fig

ure

is a

rho

mbu

s.th

en R

HO

M is

a r

hom

bus.

In r

hom

bus

AB

CD

,m!

BA

C%

32.F

ind

th

e m

easu

re o

f ea

ch n

um

bere

d a

ngl

e.A

BC

Dis

a r

hom

bus,

so t

he d

iago

nals

are

per

pend

icul

ar a

nd "

AB

Eis

a r

ight

tri

angl

e.T

hus

m!

4 "

90 a

nd m

!1

"90

!32

or

58.T

hedi

agon

als

in a

rho

mbu

s bi

sect

the

ver

tex

angl

es,s

o m

!1

"m

!2.

Thu

s,m

!2

"58

.

A r

hom

bus

is a

par

alle

logr

am,s

o th

e op

posi

te s

ides

are

par

alle

l.!

BA

Can

d !

3 ar

eal

tern

ate

inte

rior

ang

les

for

para

llel l

ines

,so

m!

3 "

32.

AB

CD

is a

rh

ombu

s.

1.If

m!

AB

D"

60,f

ind

m!

BD

C.

602.

If A

E"

8,fi

nd A

C.

16

3.If

AB

"26

and

BD

"20

,fin

d A

E.

244.

Fin

d m

!C

EB

.90

5.If

m!

CB

D"

58,f

ind

m!

AC

B.

326.

If A

E"

3x!

1 an

d A

C"

16,f

ind

x.3

7.If

m!

CD

B"

6yan

d m

!A

CB

"2y

#10

,fin

d y.

10

8.If

AD

"2x

#4

and

CD

"4x

!4,

find

x.

4

9.a.

Wha

t is

the

mid

poin

t of

F"H"

?(5

,5)

b.W

hat

is t

he m

idpo

int

of G"

J"?

(5,5

)c.

Wha

t ki

nd o

f fi

gure

is F

GH

J?

Exp

lain

.FG

HJ

is a

par

alle

logr

am b

ecau

se

the

diag

onal

s bi

sect

eac

h ot

her.

d.W

hat

is t

he s

lope

of F"

H"?

$1 2$

e.W

hat

is t

he s

lope

of G"

J"?

!2

f.B

ased

on

part

s c,

d,an

d e,

wha

t ki

nd o

f fi

gure

is F

GH

J?

Exp

lain

.FG

HJ

is a

par

alle

logr

am w

ith p

erpe

ndic

ular

dia

gona

ls,

so it

is a

rho

mbu

s.

x

y

O

J

GF

HC

AD

EB

CA

DEB

32"

12

34

MO

HR

B

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill44

2G

lenc

oe G

eom

etry

Pro

per

ties

of

Squ

ares

A s

quar

e ha

s al

l the

pro

pert

ies

of a

rho

mbu

s an

d al

l the

prop

erti

es o

f a

rect

angl

e.

Fin

d t

he

mea

sure

of

each

nu

mbe

red

an

gle

of

squ

are

AB

CD

.

Usi

ng p

rope

rtie

s of

rho

mbi

and

rec

tang

les,

the

diag

onal

s ar

e pe

rpen

dicu

lar

and

cong

ruen

t."

AB

Eis

a r

ight

tri

angl

e,so

m!

1 "

m!

2 "

90.

Eac

h ve

rtex

ang

le is

a r

ight

ang

le a

nd t

he d

iago

nals

bis

ect

the

vert

ex a

ngle

s,so

m!

3 "

m!

4 "

m!

5 "

45.

Det

erm

ine

wh

eth

er t

he

give

n v

erti

ces

rep

rese

nt

a pa

rall

elog

ram

,rec

tan

gle,

rhom

bus,

or s

qua

re.E

xpla

in y

our

reas

onin

g.

1.A

(0,2

),B

(2,4

),C

(4,2

),D

(2,0

)

Squ

are;

the

four

sid

es a

re #

and

cons

ecut

ive

side

s ar

e ⊥.

2.D

(!2,

1),E

(!1,

3),F

(3,1

),G

(2,!

1)

Rec

tang

le;b

oth

pair

s of

opp

osite

sid

es a

re ||

and

cons

ecut

ive

side

s ar

e ⊥.

3.A

(!2,

!1)

,B(0

,2),

C(2

,!1)

,D(0

,!4)

Rho

mbu

s;th

e fo

ur s

ides

are

#an

d co

nsec

utiv

e si

des

are

not

⊥.

4.A

(!3,

0),B

(!1,

3),C

(5,!

1),D

(3,!

4)

Rec

tang

le;b

oth

pair

s of

opp

osite

sid

es a

re ||

and

cons

ecut

ive

side

s ar

e ⊥.

5.S

(!1,

4),T

(3,2

),U

(1,!

2),V

(!3,

0)

Squ

are;

the

four

sid

es a

re #

and

cons

ecut

ive

side

s ar

e ⊥.

6.F

(!1,

0),G

(1,3

),H

(4,1

),I(

2,!

2)

Squ

are;

the

four

sid

es a

re #

and

cons

ecut

ive

side

s ar

e ⊥.

7.Sq

uare

RS

TU

has

vert

ices

R(!

3,!

1),S

(!1,

2),a

nd T

(2,0

).F

ind

the

coor

dina

tes

ofve

rtex

U. (

0,!

3)

C

AD

12

34

5

EB

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Rho

mbi

and

Squ

ares

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-5

8-5

Exer

cises

Exer

cises

Exam

ple

Exam

ple



An

swer

s

Skil

ls P

ract

ice

Rho

mbi

and

Squ

ares

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-5

8-5

©G

lenc

oe/M

cGra

w-H

ill44

3G

lenc

oe G

eom

etry

Lesson 8-5

Use

rh

ombu

s D

KL

Mw

ith

AM

%4x

,AK

%5x

!3,

and

D

L%

10.

1.F

ind

x.2.

Fin

d A

L.

35

3.F

ind

m!

KA

L.

4.F

ind

DM

.90

13

Use

rh

ombu

s R

ST

Vw

ith

RS

%5y

#2,

ST

%3y

#6,

and

N

V%

6.

5.F

ind

y.6.

Fin

d T

V.

212

7.F

ind

m!

NT

V.

8.F

ind

m!

SV

T.

3060

9.F

ind

m!

RS

T.

10.F

ind

m!

SR

V.

120

60

CO

OR

DIN

ATE

GEO

MET

RYG

iven

eac

h s

et o

f ve

rtic

es,d

eter

min

e w

het

her

"Q

RS

Tis

a r

hom

bus,

a re

cta

ngl

e,or

a s

qua

re.L

ist

all

that

ap

ply

.Exp

lain

you

r re

ason

ing.

11.Q

(3,5

),R

(3,1

),S

(!1,

1),T

(!1,

5)R

hom

bus,

rect

angl

e,sq

uare

;all

side

s ar

e co

ngru

ent

and

the

diag

onal

sar

e pe

rpen

dicu

lar

and

cong

ruen

t.

12.Q

(!5,

12),

R(5

,12)

,S(!

1,4)

,T(!

11,4

)R

hom

bus;

all s

ides

are

con

grue

nt a

nd t

he d

iago

nals

are

per

pend

icul

ar,

but

not

cong

ruen

t.

13.Q

(!6,

!1)

,R(4

,!6)

,S(2

,5),

T(!

8,10

)R

hom

bus;

all s

ides

are

con

grue

nt a

nd t

he d

iago

nals

are

per

pend

icul

ar,

but

not

cong

ruen

t.

14.Q

(2,!

4),R

(!6,

!8)

,S(!

10,2

),T

(!2,

6)N

one;

oppo

site

sid

es a

re c

ongr

uent

,but

the

dia

gona

ls a

re n

eith

erco

ngru

ent

nor

perp

endi

cula

r.

RV

NS

T

ML

AD

K

©G

lenc

oe/M

cGra

w-H

ill44

4G

lenc

oe G

eom

etry

Use

rh

ombu

s P

RY

Zw

ith

RK

%4y

#1,

ZK

%7y

!14

,P

K%

3x!

1,an

d Y

K%

2x#

6.

1.F

ind

PY

.2.

Fin

d R

Z.

4042

3.F

ind

RY

.4.

Fin

d m

!Y

KZ

.29

90

Use

rh

ombu

s M

NP

Qw

ith

PQ

%3!

2",P

A%

4x!

1,an

d

AM

%9x

!6.

5.F

ind

AQ

.6.

Fin

d m

!A

PQ

.3

45

7.F

ind

m!

MN

P.

8.F

ind

PM

.90

6

CO

OR

DIN

ATE

GEO

MET

RYG

iven

eac

h s

et o

f ve

rtic

es,d

eter

min

e w

het

her

"B

EF

Gis

a r

hom

bus,

a re

cta

ngl

e,or

a s

qua

re.L

ist

all

that

ap

ply

.Exp

lain

you

r re

ason

ing.

9.B

(!9,

1),E

(2,3

),F

(12,

!2)

,G(1

,!4)

Rho

mbu

s;al

l sid

es a

re c

ongr

uent

and

the

dia

gona

ls a

re p

erpe

ndic

ular

,bu

t no

t co

ngru

ent.

10.B

(1,3

),E

(7,!

3),F

(1,!

9),G

(!5,

!3)

Rho

mbu

s,re

ctan

gle,

squa

re;a

ll si

des

are

cong

ruen

t an

d th

e di

agon

als

are

perp

endi

cula

r an

d co

ngru

ent.

11.B

(!4,

!5)

,E(1

,!5)

,F(!

7,!

1),G

(!2,

!1)

Non

e;tw

o of

the

opp

osite

sid

es a

re n

ot c

ongr

uent

.

12.T

ESSE

LATI

ON

ST

he f

igur

e is

an

exam

ple

of a

tes

sella

tion

.Use

a r

uler

or

pro

trac

tor

to m

easu

re t

he s

hape

s an

d th

en n

ame

the

quad

rila

tera

ls

used

to

form

the

fig

ure.

The

figur

e co

nsis

ts o

f 6

cong

ruen

t rh

ombi

.

AN M

QP

K

P

YR

Z

Pra

ctic

e (A

vera

ge)

Rho

mbi

and

Squ

ares

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-5

8-5



Rea

din

g t

o L

earn

Math

emati

csR

hom

bi a

nd S

quar

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-5

8-5

©G

lenc

oe/M

cGra

w-H

ill44

5G

lenc

oe G

eom

etry

Lesson 8-5

Pre-

Act

ivit

yH

ow c

an y

ou r

ide

a bi

cycl

e w

ith

squ

are

wh

eels

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

5 at

the

top

of

page

431

in y

our

text

book

.

If y

ou d

raw

a d

iago

nal o

n th

e su

rfac

e of

one

of

the

squa

re w

heel

s sh

own

inth

e pi

ctur

e in

you

r te

xtbo

ok,h

ow c

an y

ou d

escr

ibe

the

two

tria

ngle

s th

at a

refo

rmed

?S

ampl

e an

swer

:tw

o co

ngru

ent

isos

cele

s ri

ght

tria

ngle

s

Rea

din

g t

he

Less

on

1.Sk

etch

eac

h of

the

fol

low

ing.

Sam

ple

answ

ers

are

give

n.a.

a qu

adri

late

ral w

ith

perp

endi

cula

r b.

a qu

adri

late

ral w

ith

cong

ruen

t di

agon

als

that

is n

ot a

rho

mbu

sdi

agon

als

that

is n

ot a

rec

tang

le

c.a

quad

rila

tera

l who

se d

iago

nals

are

pe

rpen

dicu

lar

and

bise

ct e

ach

othe

r,bu

t ar

e no

t co

ngru

ent

2.L

ist

all o

f th

e fo

llow

ing

spec

ial q

uadr

ilate

rals

tha

t ha

ve e

ach

liste

d pr

oper

ty:

para

llel

ogra

m,r

ecta

ngle

,rho

mbu

s,sq

uare

.

a.T

he d

iago

nals

are

con

grue

nt.

rect

angl

e,sq

uare

b.O

ppos

ite

side

s ar

e co

ngru

ent.

para

llelo

gram

,rec

tang

le,r

hom

bus,

squa

rec.

The

dia

gona

ls a

re p

erpe

ndic

ular

.rh

ombu

s,sq

uare

d.C

onse

cuti

ve a

ngle

s ar

e su

pple

men

tary

.pa

ralle

logr

am,r

ecta

ngle

,rho

mbu

s,sq

uare

e.T

he q

uadr

ilate

ral i

s eq

uila

tera

l.rh

ombu

s,sq

uare

f.T

he q

uadr

ilate

ral i

s eq

uian

gula

r.re

ctan

gle,

squa

reg.

The

dia

gona

ls a

re p

erpe

ndic

ular

and

con

grue

nt.

squa

reh

.A

pai

r of

opp

osit

e si

des

is b

oth

para

llel a

nd c

ongr

uent

.pa

ralle

logr

am,r

ecta

ngle

,rh

ombu

s,sq

uare

3.W

hat

is t

he c

omm

on n

ame

for

a re

gula

r qu

adri

late

ral?

Exp

lain

you

r an

swer

.S

quar

e;sa

mpl

e an

swer

:All

four

sid

es o

f a

squa

re a

re c

ongr

uent

and

all

four

angl

es a

re c

ongr

uent

.

Hel

pin

g Y

ou

Rem

emb

er

4.A

goo

d w

ay t

o re

mem

ber

som

ethi

ng is

to

expl

ain

it t

o so

meo

ne e

lse.

Supp

ose

that

you

rcl

assm

ate

Lui

s is

hav

ing

trou

ble

rem

embe

ring

whi

ch o

f th

e pr

oper

ties

he

has

lear

ned

inth

is c

hapt

er a

pply

to

squa

res.

How

can

you

hel

p hi

m?

Sam

ple

answ

er:A

squ

are

isa

para

llelo

gram

tha

t is

bot

h a

rect

angl

e an

d a

rhom

bus,

so a

ll pr

oper

ties

of p

aral

lelo

gram

s,re

ctan

gles

,and

rho

mbi

app

ly t

o sq

uare

s.

©G

lenc

oe/M

cGra

w-H

ill44

6G

lenc

oe G

eom

etry

Cre

atin

g P

ytha

gore

an P

uzzl

esB

y dr

awin

g tw

o sq

uare

s an

d cu

ttin

g th

em in

a c

erta

in w

ay,y

ouca

n m

ake

a pu

zzle

tha

t de

mon

stra

tes

the

Pyt

hago

rean

The

orem

.A

sam

ple

puzz

le is

sho

wn.

You

can

crea

te y

our

own

puzz

le b

yfo

llow

ing

the

inst

ruct

ions

bel

ow.

See

stu

dent

s’w

ork.

A s

ampl

e an

swer

is s

how

n.1.

Car

eful

ly c

onst

ruct

a s

quar

e an

d la

bel t

he le

ngth

of

a s

ide

as a

.The

n co

nstr

uct

a sm

alle

r sq

uare

to

the

righ

t of

it a

nd la

bel t

he le

ngth

of

a si

de a

s b,

as s

how

n in

the

fig

ure

abov

e.T

he b

ases

sho

uld

be a

djac

ent

and

colli

near

.

2.M

ark

a po

int

Xth

at is

bun

its

from

the

left

edg

e of

the

larg

er s

quar

e.T

hen

draw

the

seg

men

ts

from

the

upp

er le

ft c

orne

r of

the

larg

er s

quar

e to

po

int

X,a

nd f

rom

poi

nt X

to t

he u

pper

rig

ht c

orne

r of

the

sm

alle

r sq

uare

.

3.C

ut o

ut a

nd r

earr

ange

you

r fi

ve p

iece

s to

for

m a

la

rger

squ

are.

Dra

w a

dia

gram

to

show

you

r an

swer

.

4.V

erif

y th

at t

he le

ngth

of

each

sid

e is

equ

al t

o #

a2#

b"

2 ".

a

a

bX

b

b

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-5

8-5



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Trap

ezoi

ds

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill44

7G

lenc

oe G

eom

etry

Lesson 8-6

Pro

per

ties

of

Trap

ezo

ids

A t

rap

ezoi

dis

a q

uadr

ilate

ral w

ith

exac

tly

one

pair

of

para

llel s

ides

.The

par

alle

l sid

es a

re c

alle

d ba

ses

and

the

nonp

aral

lel s

ides

are

cal

led

legs

.If

the

legs

are

con

grue

nt,

the

trap

ezoi

d is

an

isos

cele

s tr

apez

oid

.In

an is

osce

les

trap

ezoi

d bo

th p

airs

of b

ase

angl

esar

e co

ngru

ent.

STU

Ris

an

isos

cele

s tr

apez

oid.

S"R"!

T"U"; !

R!

!U

, !S

!!

T

Th

e ve

rtic

es o

f A

BC

Dar

e A

(!3,

!1)

,B(!

1,3)

,C

(2,3

),an

d D

(4,!

1).V

erif

y th

at A

BC

Dis

a t

rap

ezoi

d.

slop

e of

A "B"

"% !3 1! !

(! (!1 3) )

%"

%4 2%"

2

slop

e of

A "D"

"%! 41 !!

(!(!31 ))

%"

%0 7%"

0

slop

e of

B "C"

"% 2

3 !! (!

3 1)%

"%0 3%

"0

slop

e of

C "D"

"%! 41 !!

23%

"%! 24 %

"!

2

Exa

ctly

tw

o si

des

are

para

llel,

A"D"

and

B"C"

,so

AB

CD

is a

tra

pezo

id.A

B"

CD

,so

AB

CD

isan

isos

cele

s tr

apez

oid.

In E

xerc

ises

1–3

,det

erm

ine

wh

eth

er A

BC

Dis

a t

rap

ezoi

d.I

f so

,det

erm

ine

wh

eth

er i

t is

an

iso

scel

es t

rap

ezoi

d.E

xpla

in.

1.A

(!1,

1),B

(2,1

),C

(3,!

2),a

nd D

(2,!

2)

Slo

pe o

f A"

B"%

0,sl

ope

of D"

C"%

0,sl

ope

of A"

D"%

!1,

slop

e of

B"C"

%!

3.E

xact

ly t

wo

side

s ar

e pa

ralle

l,so

AB

CD

is a

tra

pezo

id.A

D%

3!2"

and

BC

%!

10";A

D&

BC

,so

AB

CD

is n

ot is

osce

les.

2.A

(3,!

3),B

(!3,

!3)

,C(!

2,3)

,and

D(2

,3)

Slo

pe o

f A"

B"%

0,sl

ope

of D"

C"%

0,sl

ope

of B"

C"%

6,sl

ope

of A"

D"%

!6.

Exa

ctly

2 s

ides

are

|| ,so

AB

CD

is a

tra

pezo

id.B

C%

!37"

and

AD

%!

37";

BC

%A

D,s

o A

BC

Dis

isos

cele

s.3.

A(1

,!4)

,B(!

3,!

3),C

(!2,

3),a

nd D

(2,2

)

Slo

pe o

f A"

B"%

!$1 4$ ,

slop

e of

D"C"

%!

$1 4$ ,sl

ope

of B"

C"%

6,sl

ope

of A"

D"%

6.

Bot

h pa

irs

of o

ppos

ite s

ides

are

par

alle

l,so

AB

CD

is n

ot a

tra

pezo

id.

4.T

he v

erti

ces

of a

n is

osce

les

trap

ezoi

d ar

e R

(!2,

2),S

(2,2

),T

(4,!

1),a

nd U

(!4,

!1)

.V

erif

y th

at t

he d

iago

nals

are

con

grue

nt.

RT

%!

((4

!"

(!2)

)2"

#(!

"1

!2

")2

)"%

!45"

SU

%!

((!

4 !

"(2

))2

"#

(!1

"!

2)2

")"%

!45"

x

y

O

B

DA

C

T

RU

base

base

Sle

g

AB

"#

(!3

!"

(!1)

)"

2#

(!"

1 !

3"

)2 ""

#4

#1

"6"

"#

20""

2#5"

CD

"#

(2 !

4"

)2#

("

3 !

(!"

1))2

""

#4

#1

"6"

"#

20""

2#5"

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill44

8G

lenc

oe G

eom

etry

Med

ian

s o

f Tr

apez

oid

sT

he m

edia

nof

a t

rape

zoid

is t

he

segm

ent

that

join

s th

e m

idpo

ints

of

the

legs

.It

is p

aral

lel t

o th

e ba

ses,

and

its

leng

th is

one

-hal

f th

e su

m o

f th

e le

ngth

s of

the

bas

es.

In t

rape

zoid

HJK

L,M

N"

%1 2% (H

J#

LK

).

M"N"

is t

he

med

ian

of

trap

ezoi

d R

ST

U.F

ind

x.

MN

"%1 2% (

RS

#U

T)

30"

%1 2% (3x

#5

#9x

!5)

30"

%1 2% (12

x)

30"

6x5

"x

M "N"

is t

he

med

ian

of

trap

ezoi

d H

JK

L.F

ind

eac

h i

nd

icat

ed v

alu

e.

1.F

ind

MN

if H

J"

32 a

nd L

K"

60.

46

2.F

ind

LK

if H

J"

18 a

nd M

N"

28.

38

3.F

ind

MN

if H

J#

LK

"42

.21

4.F

ind

m!

LM

Nif

m!

LH

J"

116.

116

5.F

ind

m!

JKL

if H

JKL

is is

osce

les

and

m!

HL

K"

62.

62

6.F

ind

HJ

if M

N"

5x#

6,H

J"

3x#

6,an

d L

K"

8x.

24

7.F

ind

the

leng

th o

f th

e m

edia

n of

a t

rape

zoid

wit

h ve

rtic

es

A(!

2,2)

,B(3

,3),

C(7

,0),

and

D(!

3,!

2).

$3 2$ !26"

LK

NM

HJ

UT

NM

R3x

# 5

30 9x !

5S

LK

NM

HJ

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Trap

ezoi

ds

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-6

8-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Trap

ezoi

ds

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill44

9G

lenc

oe G

eom

etry

Lesson 8-6

CO

OR

DIN

ATE

GEO

MET

RYA

BC

Dis

a q

uad

rila

tera

l w

ith

ver

tice

s A

(!4,

!3)

,B

(3,!

3),C

(6,4

),D

(!7,

4).

1.V

erif

y th

at A

BC

Dis

a t

rape

zoid

.

A"B"

|| C"D"

2.D

eter

min

e w

heth

er A

BC

Dis

an

isos

cele

s tr

apez

oid.

Exp

lain

.

isos

cele

s;A

D%

!58"

and

BC

%!

58"

CO

OR

DIN

ATE

GEO

MET

RYE

FG

His

a q

uad

rila

tera

l w

ith

ver

tice

s E

(1,3

),F

(5,0

),G

(8,!

5),H

(!4,

4).

3.V

erif

y th

at E

FG

His

a t

rape

zoid

.

E"F"|| G"

H"

4.D

eter

min

e w

heth

er E

FG

His

an

isos

cele

s tr

apez

oid.

Exp

lain

.

not

isos

cele

s;E

H%

!26"

and

FG%

!34"

CO

OR

DIN

ATE

GEO

MET

RYL

MN

Pis

a q

uad

rila

tera

l w

ith

ver

tice

s L

(!1,

3),M

(!4,

1),

N(!

6,3)

,P(0

,7).

5.V

erif

y th

at L

MN

Pis

a t

rape

zoid

.L"M"

|| N"P"

6.D

eter

min

e w

heth

er L

MN

Pis

an

isos

cele

s tr

apez

oid.

Exp

lain

.

not

isos

cele

s;LP

%!

17"an

d M

N%

!8"

ALG

EBR

AF

ind

th

e m

issi

ng

mea

sure

(s)

for

the

give

n t

rap

ezoi

d.

7.Fo

r tr

apez

oid

HJK

L,S

and

Tar

e 8.

For

trap

ezoi

d W

XY

Z,P

and

Qar

em

idpo

ints

of

the

legs

.Fin

d H

J.58

mid

poin

ts o

f th

e le

gs.F

ind

WX

.5

9.Fo

r tr

apez

oid

DE

FG

,Tan

d U

are

10.F

or is

osce

les

trap

ezoi

d Q

RS

T,f

ind

the

mid

poin

ts o

f th

e le

gs.F

ind

TU

,m!

E,

leng

th o

f th

e m

edia

n,m

!Q

,and

m!

S.

and

m!

G.

28,9

5,14

542

.5,1

25,5

5

ST

RQ

125"

60 25

DE F

G

TU

85"

35"

42

14

WX

YZ

PQ

12 19

HJ

KLS

T72 86

©G

lenc

oe/M

cGra

w-H

ill45

0G

lenc

oe G

eom

etry

CO

OR

DIN

ATE

GEO

MET

RYR

ST

Uis

a q

uad

rila

tera

l w

ith

ver

tice

s R

(!3,

!3)

,S(5

,1),

T(1

0,!

2),U

(!4,

!9)

.

1.V

erif

y th

at R

ST

Uis

a t

rape

zoid

.

R"S"

|| T"U"

2.D

eter

min

e w

heth

er R

ST

Uis

an

isos

cele

s tr

apez

oid.

Exp

lain

.

not

isos

cele

s;R

U%

!37"

and

ST

%!

34"

CO

OR

DIN

ATE

GEO

MET

RYB

GH

Jis

a q

uad

rila

tera

l w

ith

ver

tice

s B

(!9,

1),G

(2,3

),H

(12,

!2)

,J(!

10,!

6).

3.V

erif

y th

at B

GH

Jis

a t

rape

zoid

.

B"G"

|| H"J"

4.D

eter

min

e w

heth

er B

GH

Jis

an

isos

cele

s tr

apez

oid.

Exp

lain

.

not

isos

cele

s;B

J%

!50"

and

GH

%!

125

"

ALG

EBR

AF

ind

th

e m

issi

ng

mea

sure

(s)

for

the

give

n t

rap

ezoi

d.

5.Fo

r tr

apez

oid

CD

EF

,Van

d Y

are

6.Fo

r tr

apez

oid

WR

LP

,Ban

d C

are

mid

poin

ts o

f th

e le

gs.F

ind

CD

.38

mid

poin

ts o

f th

e le

gs.F

ind

LP

.18

7.Fo

r tr

apez

oid

FG

HI,

Kan

d M

are

8.Fo

r is

osce

les

trap

ezoi

d T

VZ

Y,f

ind

the

mid

poin

ts o

f th

e le

gs.F

ind

FI,

m!

F,

leng

th o

f th

e m

edia

n,m

!T

,and

m!

Z.

and

m!

I.51

,40,

5519

.5,6

0,12

0

9.C

ON

STR

UC

TIO

NA

set

of

stai

rs le

adin

g to

the

ent

ranc

e of

a b

uild

ing

is d

esig

ned

in t

hesh

ape

of a

n is

osce

les

trap

ezoi

d w

ith

the

long

er b

ase

at t

he b

otto

m o

f th

e st

airs

and

the

shor

ter

base

at

the

top.

If t

he b

otto

m o

f th

e st

airs

is 2

1 fe

et w

ide

and

the

top

is 1

4,fi

ndth

e w

idth

of

the

stai

rs h

alfw

ay t

o th

e to

p.17

.5 f

t

10.D

ESK

TO

PSA

car

pent

er n

eeds

to

repl

ace

seve

ral t

rape

zoid

-sha

ped

desk

tops

in a

clas

sroo

m.T

he c

arpe

nter

kno

ws

the

leng

ths

of b

oth

base

s of

the

des

ktop

.Wha

t ot

her

mea

sure

men

ts,i

f an

y,do

es t

he c

arpe

nter

nee

d?S

ampl

e an

swer

:the

mea

sure

s of

the

bas

e an

glesV

YZ

T60

"34 5

H

KM

G

IF

140"

125"

21

36

WR

LP

BC

4266F

E

DC

VY

2818Pra

ctic

e (A

vera

ge)

Trap

ezoi

ds

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-6

8-6



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csTr

apez

oids

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-6

8-6

©G

lenc

oe/M

cGra

w-H

ill45

1G

lenc

oe G

eom

etry

Lesson 8-6

Pre-

Act

ivit

yH

ow a

re t

rap

ezoi

ds

use

d i

n a

rch

itec

ture

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

6 at

the

top

of

page

439

in y

our

text

book

.

How

mig

ht t

rape

zoid

s be

use

d in

the

inte

rior

des

ign

of a

hom

e?S

ampl

e an

swer

:flo

or t

iles

for

a ki

tche

n or

bat

hroo

m

Rea

din

g t

he

Less

on

1.In

the

fig

ure

at t

he r

ight

,EF

GH

is a

tra

pezo

id,I

is t

he

mid

poin

t of

F "E"

,and

Jis

the

mid

poin

t of

G"H"

.Ide

ntif

y ea

ch

of t

he f

ollo

win

g se

gmen

ts o

r an

gles

in t

he f

igur

e.

a.th

e ba

ses

of t

rape

zoid

EF

GH

F"G",E"

H"b.

the

two

pair

s of

bas

e an

gles

of

trap

ezoi

d E

FG

H!

Ean

d !

H;!

Fan

d !

Gc.

the

legs

of

trap

ezoi

d E

FG

HF"E"

,G"H"

d.th

e m

edia

n of

tra

pezo

id E

FG

HI"J"

2.D

eter

min

e w

heth

er e

ach

stat

emen

t is

tru

eor

fal

se.I

f th

e st

atem

ent

is f

alse

,exp

lain

why

.

a.A

tra

pezo

id is

a s

peci

al k

ind

of p

aral

lelo

gram

.Fa

lse;

sam

ple

answ

er:A

para

llelo

gram

has

tw

o pa

irs

of p

aral

lel s

ides

,but

a t

rape

zoid

has

onl

yon

e pa

ir o

f pa

ralle

l sid

es.

b.T

he d

iago

nals

of

a tr

apez

oid

are

cong

ruen

t.Fa

lse;

sam

ple

answ

er:T

his

is o

nly

true

for

isos

cele

s tr

apez

oids

.c.

The

med

ian

of a

tra

pezo

id is

par

alle

l to

the

legs

.Fa

lse;

sam

ple

answ

er:T

hem

edia

n is

par

alle

l to

the

base

s.d.

The

leng

th o

f th

e m

edia

n of

a t

rape

zoid

is t

he a

vera

ge o

f th

e le

ngth

of

the

base

s.tr

uee.

A t

rape

zoid

has

thr

ee m

edia

ns.

Fals

e;sa

mpl

e an

swer

:A t

rape

zoid

has

onl

yon

e m

edia

n.f.

The

bas

es o

f an

isos

cele

s tr

apez

oid

are

cong

ruen

t.Fa

lse:

sam

ple

answ

er:T

hele

gs a

re c

ongr

uent

.g.

An

isos

cele

s tr

apez

oid

has

two

pair

s of

con

grue

nt a

ngle

s.tr

ueh

.T

he m

edia

n of

an

isos

cele

s tr

apez

oid

divi

des

the

trap

ezoi

d in

to t

wo

smal

ler

isos

cele

str

apez

oids

.tr

ue

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

a ne

w g

eom

etri

c th

eore

m is

to

rela

te it

to

one

you

alre

ady

know

.Nam

e an

d st

ate

in w

ords

a t

heor

em a

bout

tri

angl

es t

hat

is s

imila

r to

the

the

orem

in t

his

less

on a

bout

the

med

ian

of a

tra

pezo

id.

Tria

ngle

Mid

segm

ent T

heor

em;a

mid

segm

ent

of a

tri

angl

e is

par

alle

l to

one

side

of

the

tria

ngle

,and

its

leng

th is

one

-hal

f th

e le

ngth

of

that

sid

e.

HE

JI

GF

©G

lenc

oe/M

cGra

w-H

ill45

2G

lenc

oe G

eom

etry

Qua

drila

tera

ls in

Con

stru

ctio

n

Qua

drila

tera

ls a

re o

ften

use

d in

con

stru

ctio

n w

ork.

1.T

he d

iagr

am a

t th

e ri

ght

repr

esen

ts a

roo

f fr

ame

and

show

s m

any

quad

rila

tera

ls.F

ind

the

follo

win

g sh

apes

in t

he d

iagr

am a

nd

shad

e in

the

ir e

dges

.S

ee s

tude

nts’

wor

k.

a.is

osce

les

tria

ngle

b.sc

alen

e tr

iang

le

c.re

ctan

gle

d.rh

ombu

s

e.tr

apez

oid

(not

isos

cele

s)

f.is

osce

les

trap

ezoi

d

2.T

he f

igur

e at

the

rig

ht r

epre

sent

s a

win

dow

.The

w

oode

n pa

rt b

etw

een

the

pane

s of

gla

ss is

3 in

ches

w

ide.

The

fra

me

arou

nd t

he o

uter

edg

e is

9 in

ches

w

ide.

The

out

side

mea

sure

men

ts o

f th

e fr

ame

are

60 in

ches

by

81 in

ches

.The

hei

ght

of t

he t

op a

nd

bott

om p

anes

is t

he s

ame.

The

top

thr

ee p

anes

are

th

e sa

me

size

.

a.H

ow w

ide

is t

he b

otto

m p

ane

of g

lass

?42

in.

b.H

ow w

ide

is e

ach

top

pane

of

glas

s?12

in.

c.H

ow h

igh

is e

ach

pane

of

glas

s?30

in.

3.E

ach

edge

of

this

box

has

bee

n re

info

rced

w

ith

a pi

ece

of t

ape.

The

box

is 1

0 in

ches

hi

gh,2

0 in

ches

wid

e,an

d 12

inch

es d

eep.

Wha

t is

the

leng

th o

f th

e ta

pe t

hat

has

been

use

d?16

8 in

.10

in.

12 in

.

20 in

.60 in

.

81 in

.

9 in

.

3 in

.

3 in

.

3 in

.

9 in

.

9 in

.

jack

rafte

rs

hip

rafte

r

com

mon

rafte

rs

Ro

of

Fram

e

ridge

boa

rdva

lley

rafte

r

plat

e

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-6

8-6



Stu

dy

Gu

ide

and I

nte

rven

tion

Coo

rdin

ate

Pro

of w

ith Q

uadr

ilate

rals

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-7

8-7

©G

lenc

oe/M

cGra

w-H

ill45

3G

lenc

oe G

eom

etry

Lesson 8-7

Posi

tio

n F

igu

res

Coo

rdin

ate

proo

fs u

se p

rope

rtie

s of

line

s an

d se

gmen

ts t

o pr

ove

geom

etri

c pr

oper

ties

.The

fir

st s

tep

in w

riti

ng a

coo

rdin

ate

proo

f is

to

plac

e th

e fi

gure

on

the

coor

dina

te p

lane

in a

con

veni

ent

way

.Use

the

fol

low

ing

guid

elin

es f

or p

laci

ng a

fig

ure

onth

e co

ordi

nate

pla

ne.

1.U

se th

e or

igin

as

a ve

rtex

, so

one

set o

f coo

rdin

ates

is (

0, 0

), o

r us

e th

e or

igin

as

the

cent

er o

f the

figu

re.

2.P

lace

at l

east

one

sid

e of

the

quad

rilat

eral

on

an a

xis

so y

ou w

ill h

ave

som

e ze

ro c

oord

inat

es.

3.Tr

y to

kee

p th

e qu

adril

ater

al in

the

first

qua

dran

t so

you

will

hav

e po

sitiv

e co

ordi

nate

s.

4.U

se c

oord

inat

es th

at m

ake

the

com

puta

tions

as

easy

as

poss

ible

. For

exa

mpl

e, u

se e

ven

num

bers

if y

ou

are

goin

g to

be

findi

ng m

idpo

ints

.

Pos

itio

n a

nd

lab

el a

rec

tan

gle

wit

h s

ides

aan

d b

un

its

lon

g on

th

e co

ord

inat

e p

lan

e.

•P

lace

one

ver

tex

at t

he o

rigi

n fo

r R

,so

one

vert

ex is

R(0

,0).

•P

lace

sid

e R "

U"al

ong

the

x-ax

is a

nd s

ide

R"S"

alon

g th

e y-

axis

,w

ith

the

rect

angl

e in

the

fir

st q

uadr

ant.

•T

he s

ides

are

aan

d b

unit

s,so

labe

l tw

o ve

rtic

es S

(0,a

) an

d U

(b,0

).•

Ver

tex

Tis

bun

its

righ

t an

d a

unit

s up

,so

the

four

th v

erte

x is

T(b

,a).

Nam

e th

e m

issi

ng

coor

din

ates

for

eac

h q

uad

rila

tera

l.

1.2.

3.

C(c

,c)

N(b

,c)

B(b

,c);

C(a

#b,

c)

Pos

itio

n a

nd

lab

el e

ach

qu

adri

late

ral

on t

he

coor

din

ate

pla

ne.

Sam

ple

answ

ers

are

give

n.4.

squa

re S

TU

Vw

ith

5.pa

ralle

logr

am P

QR

S6.

rect

angl

e A

BC

Dw

ith

side

sun

its

wit

h co

ngru

ent

diag

onal

sle

ngth

tw

ice

the

wid

th

x

y

D( 2

a, 0

)

C( 2

a, a

)B

( 0, a

)

A( 0

, 0)

x

y

S( a

, 0)

R( a

, b)

Q( 0

, b)

P( 0

, 0)

x

y

V( s

, 0)

U( s

, s)

T(0,

s)

S( 0

, 0)

x

y

D( a

, 0)C

( ?, c

)B

( b, ?

)

A( 0

, 0)

x

y

Q( a

, 0)

P( a

! b

, c)

N( ?

, ?)

M( 0

, 0)

x

y

B( c

, 0)

C( ?

, ?)

D( 0

, c)

A( 0

, 0)

x

y

U( b

, 0)

T( b

, a)

S( 0

, a)

R( 0

, 0)

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill45

4G

lenc

oe G

eom

etry

Pro

ve T

heo

rem

sA

fter

a f

igur

e ha

s be

en p

lace

d on

the

coo

rdin

ate

plan

e an

d la

bele

d,a

coor

dina

te p

roof

can

be

used

to

prov

e a

theo

rem

or

veri

fy a

pro

pert

y.T

he D

ista

nce

Form

ula,

the

Slop

e Fo

rmul

a,an

d th

e M

idpo

int

The

orem

are

oft

en u

sed

in a

coo

rdin

ate

proo

f.

Wri

te a

coo

rdin

ate

pro

of t

o sh

ow t

hat

th

e d

iago

nal

s of

a s

quar

e ar

e p

erp

end

icu

lar.

The

fir

st s

tep

is t

o po

siti

on a

nd la

bel a

squ

are

on t

he c

oord

inat

e pl

ane.

Pla

ce it

in t

he f

irst

qua

dran

t,w

ith

one

side

on

each

axi

s.L

abel

the

ver

tice

s an

d dr

aw t

he d

iago

nals

.

Giv

en:s

quar

e R

ST

UP

rove

:S "U"

⊥R"

T"P

roof

:The

slo

pe o

f S"U"

is %0 a

! !a 0

%"

!1,

and

the

slop

e of

R"T"

is %a a

! !0 0

%"

1.

The

pro

duct

of

the

two

slop

es is

!1,

so S"

U"⊥

R"T"

.

Wri

te a

coo

rdin

ate

pro

of t

o sh

ow t

hat

th

e le

ngt

h o

f th

e m

edia

n o

f a

trap

ezoi

d i

sh

alf

the

sum

of

the

len

gth

s of

th

e ba

ses.

Pos

ition

A"D"

on t

he x

-axi

s an

d B"

C"pa

ralle

l to

A"D"

.La

bel t

wo

vert

ices

A(0

,0)

and

D(2

a,0)

.Lab

el a

noth

erve

rtex

B(2

b,2c

).Fo

r ve

rtex

C,t

he y

valu

e is

the

sam

e as

for

vert

ex B

,so

the

last

ver

tex

is C

(2d,

2c).

The

coor

dina

tes

of m

idpo

int

Mar

e

$$2b2#

0$

,$0

# 22c $

%%(b

,c);

the

coor

dina

tes

of m

idpo

int

Nar

e

$$2a# 2

2d$

,$2c

2#0

$%%

(a#

d,c)

.B"C"

,M"N"

,and

A"D"

are

hori

zont

al s

egm

ents

,

so f

ind

thei

r le

ngth

s by

sub

trac

ting

the

x-co

ordi

nate

s.

AD

%2a

!0

%2a

,BC

%2d

!2b

,MN

%a

#d

!b

Then

$1 2$ (A

D#

BC

) %

$1 2$ (2a

#2d

!2b

) %

a#

d!

b%

MN

.

Ther

efor

e,th

e le

ngth

of

the

med

ian

of a

tra

pezo

id is

hal

f th

e su

m o

f th

ele

ngth

s of

the

bas

es.

x

y

D( 2

a, 0

)

B( 2

b, 2

c)

MN

C( 2

d, 2

c)

A( 0

, 0)

x

y

U( a

, 0)

T(a,

a)

S( 0

, a)

R( 0

, 0)

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Coo

rdin

ate

Pro

of W

ith Q

uadr

ilate

rals

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-7

8-7

Exam

ple

Exam

ple

Exer

cise

Exer

cise



An

swer

s

Skil

ls P

ract

ice

Coo

rdin

ate

Pro

of w

ith Q

uadr

ilate

rals

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-7

8-7

©G

lenc

oe/M

cGra

w-H

ill45

5G

lenc

oe G

eom

etry

Lesson 8-7

Pos

itio

n a

nd

lab

el e

ach

qu

adri

late

ral

on t

he

coor

din

ate

pla

ne.

1.re

ctan

gle

wit

h le

ngth

2a

unit

s an

d 2.

isos

cele

s tr

apez

oid

wit

h he

ight

aun

its,

heig

ht a

unit

sba

ses

c!

bun

its

and

b#

cun

its

Nam

e th

e m

issi

ng

coor

din

ates

for

eac

h q

uad

rila

tera

l.

3.re

ctan

gle

C(c

,a)

4.re

ctan

gle

G(0

,2b)

5.pa

ralle

logr

amR

(a#

b,c)

,S(a

,0)

6.is

osce

les

trap

ezoi

dW

(a#

b,c)

,Y(0

,c)

Pos

itio

n a

nd

lab

el t

he

figu

re o

n t

he

coor

din

ate

pla

ne.

Th

en w

rite

a c

oord

inat

ep

roof

for

th

e fo

llow

ing.

7.T

he s

egm

ents

join

ing

the

mid

poin

ts o

f th

e si

des

of a

rho

mbu

s fo

rm a

rec

tang

le.

Giv

en:A

BC

Dis

a r

hom

bus.

M,N

,P,a

nd Q

are

the

mid

poin

ts

of A"

B",B"

C",C"

D",a

nd D"

A",r

espe

ctiv

ely.

Pro

ve:

MN

PQ

is a

rec

tang

le.

Pro

of:

M%

(!a,

b),N

%(a

,b),

P%

(a,!

b),a

nd Q

%(!

a,!

b).

slop

e of

M"N"

%$ a

b !! (!

b a)$

or 0

slop

e of

Q"P"

%$!

!ba!

!(!ab)

$or

0

slop

e of

M"Q"

%$ !

! ab !

! (!b a)

$or

und

efin

edsl

ope

of N"

P"%

$ba!

!(!ab)

$or un

defin

ed

M"N"

and

Q"P"

have

the

sam

e sl

ope.

M"Q"

and

N"P"

also

hav

e th

e sa

me

slop

e.M"

N"is

per

pend

icul

ar t

o M"

Q".T

here

fore

,bot

h pa

irs

of o

ppos

ite s

ides

are

para

llel a

nd c

onse

cutiv

e si

des

are

perp

endi

cula

r.Th

is m

eans

tha

t M

NP

Qis

a r

ecta

ngle

.

xO

y B( 0

, 2b)

D( 0

, !2b

)PN

M Q

C( 2

a, 0

)

A( !

2a, 0

)

x

y

T(b,

0)

W( ?

, c)

Y( ?

, ?)

S( a

, 0)

Ox

y

S( a

, ?)

R( ?

, ?)

P( b

, c)

T(0,

0)

O

x

y

E( 5

b, 0

)

F(5b

, 2b)

G( ?

, ?)

D( 0

, 0)

Ox

y

U( c

, 0)

C( ?

, ?)

D( 0

, a)

A( 0

, 0)

O

xO

y

B( b

# c

, 0)

B( b

, a)

C( c

, a)

A( 0

, 0)

xO

y

B( 2

a, 0

)

D( 0

, a)

C( 2

a, a

)

A( 0

, 0)

©G

lenc

oe/M

cGra

w-H

ill45

6G

lenc

oe G

eom

etry

Pos

itio

n a

nd

lab

el e

ach

qu

adri

late

ral

on t

he

coor

din

ate

pla

ne.

1.pa

ralle

logr

am w

ith

side

leng

th b

unit

s 2.

isos

cele

s tr

apez

oid

wit

h he

ight

bun

its,

and

heig

ht a

unit

sba

ses

2c!

aun

its

and

2c#

aun

its

Nam

e th

e m

issi

ng

coor

din

ates

for

eac

h q

uad

rila

tera

l.

3.pa

ralle

logr

am4.

isos

cele

s tr

apez

oid

K(2

b#

c,a)

,L(c

,a)

X(!

a,0)

,Y(!

b,c)

Pos

itio

n a

nd

lab

el t

he

figu

re o

n t

he

coor

din

ate

pla

ne.

Th

en w

rite

a c

oord

inat

ep

roof

for

th

e fo

llow

ing.

5.T

he o

ppos

ite

side

s of

a p

aral

lelo

gram

are

con

grue

nt.

Giv

en:A

BC

Dis

a p

aral

lelo

gram

.P

rove

:A"B"

#C"

D",A"

D"#

B"C"

Pro

of:

AB

%!

(a!

0"

)2#

("

0 !

0"

)2 "%

!a

2 "or

a

CD

%!

[(a

#"

b) !

b"

]2#

("

c!

c"

)2 "%

!a

2 "or

a

AD

%!

(b!

0"

)2#

("

c!

0"

)2 "%

!b

2#

"c

2 "A

B%

![(

a#

"b)

!a

"]2

#(c

"!

0)2

"%

!b

2#

"c

2 "A

B%

CD

and

AD

%B

C,s

o A"

B"#

C"D"

and

A"D"

#B"

C"

6.TH

EATE

RA

sta

ge is

in t

he s

hape

of

a tr

apez

oid.

Wri

te a

co

ordi

nate

pro

of t

o pr

ove

that

T "R"

and

S"F"ar

e pa

ralle

l.G

iven

:T(1

0,0)

,R(2

0,0)

,S(3

0,25

),F

(0,2

5)P

rove

:T"R"

|| S"F"

Pro

of:T

he s

lope

of

T"R"%

$ 20 0! !

0 10$

%$ 10 0$

and

the

slop

e of

SF

%$2 35 0! !

2 05$

%$ 30 0$

.Sin

ce T"

R"an

d

S"F"bo

th h

ave

a sl

ope

of 0

,T"R"

|| S"F".

x

y T(10

, 0)

S( 3

0, 2

5)F(

0, 2

5)

R( 2

0, 0

)O

xO

y

B( a

, 0)

D( b

, c)

C( a

# b

, c)

A( 0

, 0)

x

y

W( a

, 0)

Z(b,

c)

Y( ?

, ?)

X( ?

, ?)

Ox

y

J(2b

, 0)

K( ?

, a)

L(c,

?)

H( 0

, 0)

O

xO

y

B( a

# 2

c, 0

)

D( a

, b)

C( 2

c, b

)

A( 0

, 0)

xO

y

B( b

, 0)

D( c

, a)

C( b

# c

, a)

A( 0

, 0)P

ract

ice

(Ave

rage

)

Coo

rdin

ate

Pro

of w

ith Q

uadr

ilate

rals

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-7

8-7



Rea

din

g t

o L

earn

Math

emati

csC

oord

inat

e P

roof

with

Qua

drila

tera

ls

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-7

8-7

©G

lenc

oe/M

cGra

w-H

ill45

7G

lenc

oe G

eom

etry

Lesson 8-7

Pre-

Act

ivit

yH

ow c

an y

ou u

se a

coo

rdin

ate

pla

ne

to p

rove

th

eore

ms

abou

tqu

adri

late

rals

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 8-

7 at

the

top

of

page

447

in y

our

text

book

.

Wha

t sp

ecia

l kin

ds o

f qua

drila

tera

ls c

an b

e pl

aced

on

a co

ordi

nate

sys

tem

so

that

tw

o si

des

of t

he q

uadr

ilate

ral l

ie a

long

the

axe

s?re

ctan

gles

and

squ

ares

Rea

din

g t

he

Less

on

1.F

ind

the

mis

sing

coo

rdin

ates

in e

ach

figu

re.T

hen

wri

te t

he c

oord

inat

es o

f th

e fo

urve

rtic

es o

f th

e qu

adri

late

ral.

a.is

osce

les

trap

ezoi

db.

para

llelo

gram

R(!

a,0)

,S(!

b,c)

,T(b

,c),

U(a

,0)

M(0

,0),

N(0

,a),

P(b

,c#

a),Q

(b,c

)

2.R

efer

to

quad

rila

tera

l EF

GH

.

a.F

ind

the

slop

e of

eac

h si

de.

slop

e E"F"

%3,

slop

e F"G"

%$1 3$ ,

slop

e G"

H"%

3,sl

ope

H"E"

%$1 3$

b.F

ind

the

leng

th o

f ea

ch s

ide.

EF

%!

10",F

G%

!10"

,GH

%!

10",H

E%

!10"

c.F

ind

the

slop

e of

eac

h di

agon

al.

slop

e E"G"

%1,

slop

e F"H"

%!

1d.

Fin

d th

e le

ngth

of

each

dia

gona

l.E

G%

!32"

,FH

%!

8"e.

Wha

t ca

n yo

u co

nclu

de a

bout

the

sid

es o

f EF

GH

?A

ll fo

ur s

ides

are

con

grue

ntan

d bo

th p

airs

of

oppo

site

sid

es a

re p

aral

lel.

f.W

hat

can

you

conc

lude

abo

ut t

he d

iago

nals

of E

FG

H?

The

diag

onal

s ar

epe

rpen

dicu

lar

and

they

are

not

con

grue

nt.

g.C

lass

ify

EF

GH

as a

par

alle

logr

am,a

rho

mbu

s,or

a s

quar

e.C

hoos

e th

e m

ost

spec

ific

term

.Exp

lain

how

you

r re

sult

s fr

om p

arts

a-f

sup

port

you

r co

nclu

sion

.E

FGH

is a

rho

mbu

s.A

ll fo

ur s

ides

are

con

grue

nt a

nd t

he d

iago

nals

ar

e pe

rpen

dicu

lar.

Sin

ce t

he d

iago

nals

are

not

con

grue

nt,E

FGH

is n

ota

squa

re.

Hel

pin

g Y

ou

Rem

emb

er3.

Wha

t is

an

easy

way

to

rem

embe

r ho

w b

est

to d

raw

a d

iagr

am t

hat

will

hel

p yo

u de

vise

a co

ordi

nate

pro

of?

Sam

ple

answ

er:A

key

poi

nt in

the

coo

rdin

ate

plan

e is

the

orig

in.T

he e

very

day

mea

ning

of

orig

inis

pla

ce w

here

som

ethi

ngbe

gins

.So

look

to

see

if th

ere

is a

goo

d w

ay t

o be

gin

by p

laci

ng a

ver

tex

of t

he f

igur

e at

the

ori

gin.

x

y

O

E

FG

H

x

y

Q( b

, ?)

P( ?

, c #

a)

N( ?

, a)

M( ?

, ?)

Ox

y

U( a

, ?)

T( b

, ?)

S( ?

, c)

R( ?

, ?)

O

©G

lenc

oe/M

cGra

w-H

ill45

8G

lenc

oe G

eom

etry

Coo

rdin

ate

Pro

ofs

An

impo

rtan

t pa

rt o

f pl

anni

ng a

coo

rdin

ate

proo

f is

cor

rect

ly p

laci

ng a

nd la

belin

g th

ege

omet

ric

figu

re o

n th

e co

ordi

nate

pla

ne.

Dra

w a

dia

gram

you

wou

ld u

se i

n a

coo

rdin

ate

pro

of o

f th

efo

llow

ing

theo

rem

.

The

med

ian

of a

tra

pezo

id i

s pa

rall

el t

o th

e ba

ses

of t

he t

rape

zoid

.

In t

he d

iagr

am,n

ote

that

one

ver

tex,

A,i

s pl

aced

at

the

orig

in.A

lso,

the

coor

dina

tes

of B

,C,a

nd D

use

2a,2

b,an

d 2c

in o

rder

to

avoi

d th

e us

e of

fra

ctio

ns w

hen

find

ing

the

coor

dina

tes

of t

he m

idpo

ints

,M

and

N.

Whe

n do

ing

coor

dina

te p

roof

s,th

e fo

llow

ing

stra

tegi

es m

ay b

e he

lpfu

l.

1.If

you

are

ask

ed t

o pr

ove

that

seg

men

ts a

re p

aral

lel o

r pe

rpen

dicu

lar,

use

slop

es.

2.If

you

are

ask

ed t

o pr

ove

that

seg

men

ts a

re c

ongr

uent

or

have

rel

ated

mea

sure

s,us

e th

e di

stan

ce f

orm

ula.

3.If

you

are

ask

ed t

o pr

ove

that

a s

egm

ent

is b

isec

ted,

use

the

mid

poin

t fo

rmul

a.

For

eac

h o

f th

e fo

llow

ing

theo

rem

s,a

dia

gram

has

bee

n p

rovi

ded

to

be u

sed

in

afo

rmal

pro

of.N

ame

the

mis

sin

g co

ord

inat

es i

n t

he

dia

gram

.Th

en,u

sin

g th

e G

iven

and

th

e P

rove

stat

emen

ts,p

rove

th

e th

eore

m.

1.T

he m

edia

n of

a t

rape

zoid

is p

aral

lel t

o th

e ba

ses

of t

he t

rape

zoid

.(U

se t

he d

iagr

amgi

ven

in t

he e

xam

ple

abov

e.)

Coo

rdin

ates

of M

and

N:

M(b

,c);

N(d

#a,

c)

Giv

en:T

rape

zoid

AB

CD

has

med

ian

M"N"

.P

rove

:B "C"

|| M"N"

and

A"D"

|| M"N"

Pro

of:

The

slop

e of

A"D"

is $ 20 a! !

0 0$

or 0

.The

slo

pe o

f B"

C"is

$ 22 dc! !

2 2c b$

or 0

.

The

slop

e of

M"N"

is $ (a

#c! b)

c !b

$or

0.S

ince

all

thre

e sl

opes

are

equa

l,B"

C"|| M"

N"an

d A"

D"|| M"

N".

2.T

he m

edia

ns t

o th

e le

gs o

f an

isos

cele

s tr

iang

le a

re c

ongr

uent

.

Coo

rdin

ates

of T

and

K:T

(a,b

);K

(3a,

b)

Giv

en:"

AB

Cis

isos

cele

s w

ith

med

ians

T"B"

and

K"A"

.P

rove

:T "B"

!K"

A"

Pro

of:

TB%

!(4

a!

"a)

2#

"(0

!"

b)2

"or

!9a

2#

"b

2"

KA

%!

(3a

!"

0)2

#"

(b!

"0)

2"

or !

9a2

#"

b2

"S

ince

TB

%K

A,T"

B"#

K"A"

.

x

y

O

TK

A( 0

, 0)

B( 4

a, 0

)

C( 2

a, 2

b)

x

y

O

MN

A( 0

, 0)

D( 2

a, 0

)

C( 2

d, 2

c)B

( 2b,

2c)

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

8-7

8-7

Exam

ple

Exam

ple


chapter 8 resource masters - · pdf filechapter 8 test, form 2d . . . . . . . . . . ....

Documents