chapter 6 resource masters - math problem solving workbook 0-07-860193-2 reading to learn...
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Chapter 6Resource 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 6 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: 0-07-860183-5 GeometryChapter 6 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 6-1Study Guide and Intervention . . . . . . . . 295–296Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 297Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 298Reading to Learn Mathematics . . . . . . . . . . 299Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 300
Lesson 6-2Study Guide and Intervention . . . . . . . . 301–302Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 303Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 304Reading to Learn Mathematics . . . . . . . . . . 305Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 306
Lesson 6-3Study Guide and Intervention . . . . . . . . 307–308Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 309Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 310Reading to Learn Mathematics . . . . . . . . . . 311Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 312
Lesson 6-4Study Guide and Intervention . . . . . . . . 313–314Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 315Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 316Reading to Learn Mathematics . . . . . . . . . . 317Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 318
Lesson 6-5Study Guide and Intervention . . . . . . . . 319–320Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 321Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 322Reading to Learn Mathematics . . . . . . . . . . 323Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 324
Lesson 6-6Study Guide and Intervention . . . . . . . . 325–326Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 327Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 328Reading to Learn Mathematics . . . . . . . . . . 329Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 330
Chapter 6 AssessmentChapter 6 Test, Form 1 . . . . . . . . . . . . 331–332Chapter 6 Test, Form 2A . . . . . . . . . . . 333–334Chapter 6 Test, Form 2B . . . . . . . . . . . 335–336Chapter 6 Test, Form 2C . . . . . . . . . . . 337–338Chapter 6 Test, Form 2D . . . . . . . . . . . 339–340Chapter 6 Test, Form 3 . . . . . . . . . . . . 341–342Chapter 6 Open-Ended Assessment . . . . . . 343Chapter 6 Vocabulary Test/Review . . . . . . . 344Chapter 6 Quizzes 1 & 2 . . . . . . . . . . . . . . . 345Chapter 6 Quizzes 3 & 4 . . . . . . . . . . . . . . . 346Chapter 6 Mid-Chapter Test . . . . . . . . . . . . 347Chapter 6 Cumulative Review . . . . . . . . . . . 348Chapter 6 Standardized Test Practice . 349–350
Standardized Test Practice Student Recording Sheet . . . . . . . . . . . . . . A1
ANSWERS . . . . . . . . . . . . . . . . . . . . . . A2–A29
© Glencoe/McGraw-Hill iv Glencoe Geometry
Teacher’s Guide to Using theChapter 6 Resource Masters
The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 6 Resource Masters includes the core materials neededfor Chapter 6. These materials include worksheets, extensions, and assessment options.The answers for these pages appear at the back of this booklet.
All of the materials found in this booklet are included for viewing and printing 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 6-1.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 6-1.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 6Resources 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 338–339. 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 _____
66
© 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 6.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
cross products
extremes
fractal
iteration
ID·uh·RAY·shuhn
means
(continued on the next page)
© Glencoe/McGraw-Hill viii Glencoe Geometry
Vocabulary Term Found on Page Definition/Description/Example
midsegment
proportion
ratio
scale factor
self-similar
similar polygons
Reading to Learn MathematicsVocabulary Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
66
Learning to Read MathematicsProof Builder
NAME ______________________________________________ DATE ____________ PERIOD _____
66
© Glencoe/McGraw-Hill ix Glencoe Geometry
Proo
f Bu
ilderThis is a list of key theorems and postulates you will learn in Chapter 6. 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 6.1Side-Side-Side (SSS) Similarity
Theorem 6.2Side-Angle-Side (SAS) Similarity
Theorem 6.3
Theorem 6.4Triangle Proportionality Theorem
Theorem 6.5Converse of the TriangleProportionality Theorem
Theorem 6.6Triangle Midsegment Theorem
(continued on the next page)
© Glencoe/McGraw-Hill x Glencoe Geometry
Theorem or Postulate Found on Page Description/Illustration/Abbreviation
Theorem 6.7Proportional Perimeters Theorem
Theorem 6.8
Theorem 6.9
Theorem 6.10
Theorem 6.11Angle Bisector Theorem
Postulate 6.1Angle-Angle (AA) Similarity
Learning to Read MathematicsProof Builder (continued)
NAME ______________________________________________ DATE ____________ PERIOD _____
66
Study Guide and InterventionProportions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
© Glencoe/McGraw-Hill 295 Glencoe Geometry
Less
on
6-1
Write Ratios A ratio is a comparison of two quantities. The ratio a to b, where b is not zero, can be written as �
ab� or a:b. The ratio of two quantities is sometimes called a scale
factor. For a scale factor, the units for each quantity are the same.
In 2002, the Chicago Cubs baseball team won 67 games out of 162.Write a ratio for the number of games won to the total number of games played.To find the ratio, divide the number of games won by the total number of games played. The
result is �16672�, which is about 0.41. The Chicago Cubs won about 41% of their games in 2002.
A doll house that is 15 inches tall is a scale model of a real housewith a height of 20 feet. What is the ratio of the height of the doll house to theheight of the real house?To start, convert the height of the real house to inches.20 feet � 12 inches per foot � 240 inchesTo find the ratio or scale factor of the heights, divide the height of the doll house by theheight of the real house. The ratio is 15 inches:240 inches or 1:16. The height of the doll
house is �116� the height of the real house.
1. In the 2002 Major League baseball season, Sammy Sosa hit 49 home runs and was atbat 556 times. Find the ratio of home runs to the number of times he was at bat.
2. There are 182 girls in the sophomore class of 305 students. Find the ratio of girls to total students.
3. The length of a rectangle is 8 inches and its width is 5 inches. Find the ratio of length to width.
4. The sides of a triangle are 3 inches, 4 inches, and 5 inches. Find the scale factor betweenthe longest and the shortest sides.
5. The length of a model train is 18 inches. It is a scale model of a train that is 48 feet long.Find the scale factor.
Example 1Example 1
Example 2Example 2
ExercisesExercises
© Glencoe/McGraw-Hill 296 Glencoe Geometry
Use Properties of Proportions A statement that two ratios are equal is called aproportion. In the proportion �
ab� � �d
c�, where b and d are not zero, the values a and d are
the extremes and the values b and c are the means. In a proportion, the product of themeans is equal to the product of the extremes, so ad � bc.
�ab� � �d
c�
a � d � b � c↑ ↑
extremes means
Solve �196� � �
2x7�.
�196� � �
2x7�
9 � x � 16 � 27 Cross products
9x � 432 Multiply.
x � 48 Divide each side by 9.
A room is 49 centimeters by 28 centimeters on a scale drawing of ahouse. For the actual room, the larger dimension is 14 feet. Find the shorterdimension of the actual room.If x is the room’s shorter dimension, then
�2489� � �1
x4�
49x � 392 Cross products
x � 8 Divide each side by 49.
The shorter side of the room is 8 feet.
Solve each proportion.
1. �12� � �
2x8� 2. �
38� � �2
y4� 3. �
xx
��
222
� � �3100�
4. �183.2� � �
9y� 5. �
2x8� 3� � �
54� 6. �
xx
��
11� � �
34�
Use a proportion to solve each problem.
7. If 3 cassettes cost $44.85, find the cost of one cassette.
8. The ratio of the sides of a triangle are 8:15:17. If the perimeter of the triangle is 480 inches, find the length of each side of the triangle.
9. The scale on a map indicates that one inch equals 4 miles. If two towns are 3.5 inchesapart on the map, what is the actual distance between the towns?
shorter dimension���longer dimension
Study Guide and Intervention (continued)
Proportions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
Example 1Example 1
Example 2Example 2
ExercisesExercises
Skills PracticeProportions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
© Glencoe/McGraw-Hill 297 Glencoe Geometry
Less
on
6-1
1. FOOTBALL A tight end scored 6 touchdowns in 14 games. Find the ratio of touchdownsper game.
2. EDUCATION In a schedule of 6 classes, Marta has 2 elective classes. What is the ratioof elective to non-elective classes in Marta’s schedule?
3. BIOLOGY Out of 274 listed species of birds in the United States, 78 species made theendangered list. Find the ratio of endangered species of birds to listed species in theUnited States.
4. ART An artist in Portland, Oregon, makes bronze sculptures of dogs. The ratio of theheight of a sculpture to the actual height of the dog is 2:3. If the height of the sculptureis 14 inches, find the height of the dog.
5. SCHOOL The ratio of male students to female students in the drama club at CampbellHigh School is 3:4. If the number of male students in the club is 18, what is the numberof female students?
Solve each proportion.
6. �25� � �4
x0� 7. �1
70� � �
2x1� 8. �
250� � �
46x�
9. �54x� � �
385� 10. �
x �3
1� � �
72� 11. �
135� � �
x �5
3�
Find the measures of the sides of each triangle.
12. The ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is 450 centimeters.
13. The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 220 meters.
14. The ratio of the measures of the sides of a triangle is 4:6:8, and its perimeter is 126 feet.
15. The ratio of the measures of the sides of a triangle is 5:7:8, and its perimeter is 40 inches.
© Glencoe/McGraw-Hill 298 Glencoe Geometry
1. NUTRITION One ounce of cheddar cheese contains 9 grams of fat. Six of the grams of fatare saturated fats. Find the ratio of saturated fats to total fat in an ounce of cheese.
2. FARMING The ratio of goats to sheep at a university research farm is 4:7. The numberof sheep at the farm is 28. What is the number of goats?
3. ART Edward Hopper’s oil on canvas painting Nighthawks has a length of 60 inches anda width of 30 inches. A print of the original has a length of 2.5 inches. What is the widthof the print?
Solve each proportion.
4. �58� � �1
x2� 5. �1.
x12� � �
15� 6. �2
67x� � �
43�
7. �x �
32
� � �89� 8. �
3x4� 5� � �
�75� 9. �
x �4
2� � �
x �2
4�
Find the measures of the sides of each triangle.
10. The ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104 feet.
11. The ratio of the measures of the sides of a triangle is 7:9:12, and its perimeter is 84 inches.
12. The ratio of the measures of the sides of a triangle is 6:7:9, and its perimeter is 77 centimeters.
Find the measures of the angles in each triangle.
13. The ratio of the measures of the angles is 4:5:6.
14. The ratio of the measures of the angles is 5:7:8.
15. BRIDGES The span of the Benjamin Franklin suspension bridge in Philadelphia,Pennsylvania, is 1750 feet. A model of the bridge has a span of 42 inches. What is theratio of the span of the model to the span of the actual Benjamin Franklin Bridge?
Practice Proportions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
Reading to Learn MathematicsProportions
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
© Glencoe/McGraw-Hill 299 Glencoe Geometry
Less
on
6-1
Pre-Activity How do artists use ratios?
Read the introduction to Lesson 6-1 at the top of page 282 in your textbook.
Estimate the ratio of length to width for the background rectangles inTiffany’s Clematis Skylight.
Reading the Lesson
1. Match each description in the first column with a word or phrase from the secondcolumn.
a. The ratio of two corresponding quantities i. proportion
b. r and u in the equation �rs� � �u
t� ii. cross products
c. a comparison of two quantities iii. means
d. ru and st in the equation �rs� � �u
t� iv. scale factor
e. an equation stating that two ratios are equal v. extremes
f. s and t in the equation �rs� � �u
t� vi. ratio
2. If m, n, p, and q are nonzero numbers such that �mn� � �
pq�, which of the following
statements could be false?
A. np � mq B. �np
� � �mq�
C. mp � nq D. qm � pn
E. �mn� � �p
q� F. �p
q� � �m
n�
G. m:p � n:q H. m:n � p:q
3. Write two proportions that match each description.
a. Means are 5 and 8; extremes are 4 and 10.
b. Means are 5 and 4; extremes are positive integers that are different from means.
Helping You Remember
4. Sometimes it is easier to remember a mathematical idea if you put it into words withoutusing any mathematical symbols. How can you use this approach to remember theconcept of equality of cross products?
© Glencoe/McGraw-Hill 300 Glencoe Geometry
Golden Rectangles
Use a straightedge, compass, and the instructions belowto construct a golden rectangle.
1. Construct square ABCD with sides of 2 cm.
2. Construct the midpoint of A�B�. Call the midpoint M.
3. Draw AB���. Set your compass at an opening equal to MC.Use M as the center to draw an arc that intersects AB���. Call the point of intersection P.
4. Construct a line through P that is perpendicular to AB���.
5. Draw DC��� so that it intersects the perpendicular line in step 4.Call the intersection point Q. APQD is a golden rectanglebecause the ratio of its length to its width is 1.618. Check this
conclusion by finding the value of �QAPP�. Rectangles whose sides
have this ratio are, it is said, the most pleasing to the human eye.
A figure consisting of similar golden rectangles is shown below. Use a compassand the instructions below to draw quarter-circle arcs that form a spiral like thatfound in the shell of a chambered nautilus.
6. Using A as a center, draw an arc that passes through B and C.
7. Using D as a center, draw an arc that passes through C and E.
8. Using F as a center, draw an arc that passes through E and G.
9. Using H as a center, draw an arc that passes through G and J.
10. Using K as a center, draw an arc that passes through J and L.
11. Using M as a center, draw an arc that passes through L and N.
M NH
L FD
JK
B A
C
G
E
A M B P
D C Q
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-16-1
Study Guide and InterventionSimilar Polygons
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
© Glencoe/McGraw-Hill 301 Glencoe Geometry
Less
on
6-2
Identify Similar Figures
Determine whether the triangles are similar.Two polygons are similar if and only if their corresponding angles are congruent and their corresponding sides are proportional.
�C � �Z because they are right angles, and �B � �X.By the Third Angle Theorem, �A � �Y.
For the sides, �BX
CZ� � �
2203�, �
BX
AY� � � �
2203�, and �
AY
CZ� � �
2203�.
The side lengths are proportional. So �BCA � �XZY.
Is polygon WXYZ � polygon PQRS?
For the sides, �WPQ
X� � �
182� � �
32�, �Q
XYR� � �
1182� � �
32�, �R
YZS� � �
1150� � �
32�,
and �ZSWP� � �
96� � �
32�. So corresponding sides are proportional.
Also, �W � �P, �X � �Q, �Y � �R, and �Z � �S, so corresponding angles are congruent. We can conclude that polygon WXYZ � polygon PQRS.
Determine whether each pair of figures is similar. If they are similar, give theratio of corresponding sides.
1. 2.
3. 4. 10y
8y
37�
37�116�
27�
5x
4x
15z12z
22
11
1610
equilateral triangles
15
9
12
18
XW
Z Y10
6
8
12
QP
S R
20�2��23�2�
20 23
2320 20��2 23��245�
45�C A X Z
B Y
ExercisesExercises
Example 1Example 1
Example 2Example 2
© Glencoe/McGraw-Hill 302 Glencoe Geometry
Scale Factors When two polygons are similar, the ratio of the lengths of corresponding sides is called the scale factor. At the right, �ABC � �XYZ. The scale factor of �ABC to �XYZ is 2 and the scale factor of
�XYZ to �ABC is �12�.
3 cm
6 cm
8 cm
10 cm5 cm4 cm
Z
Y
XC
A
B
Study Guide and Intervention (continued)
Similar Polygons
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
The two polygons aresimilar. Find x and y.
Use the congruent angles to write thecorresponding vertices in order.�RST � �MNPWrite proportions to find x and y.
�3126� � �1
x3� �
3y8� � �
3126�
16x � 32(13) 32y � 38(16)x � 26 y � 19
x
y38
3216 13
T
S
R P
N
M
�ABC � �CDE. Find the scale factor and find the lengths of C�D�and D�E�.
AC � 3 � 0 � 3 and CE � 9 � 3 � 6. Thescale factor of �CDE to �ABC is 6:3 or 2:1.Using the Distance Formula,AB � �1 � 9� � �10� and BC � �4 � 9� � �13�. The lengths of thesides of �CDE are twice those of �ABC,so DC � 2(BA) or 2�10� and DE � 2(BC) or 2�13�.
x
y
A(0, 0)
B(1, 3)
C(3, 0)
E(9, 0)
D
Example 1Example 1 Example 2Example 2
ExercisesExercises
Each pair of polygons is similar. Find x and y.
1. 2.
3. 4.
5. In Example 2 above, point D has coordinates (5, 6). Use the Distance Formula to verifythe lengths of C�D� and D�E�.
40
36x
30
18 y12
y � 1
18
24
x18
10
y
4.5x
2.5
5
9
12
y8
10x
Skills PracticeSimilar Polygons
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
© Glencoe/McGraw-Hill 303 Glencoe Geometry
Less
on
6-2
Determine whether each pair of figures is similar. Justify your answer.
1. 2.
Each pair of polygons is similar. Write a similarity statement, and find x, themeasure(s) of the indicated side(s), and the scale factor.
3. G�H� 4. S�T� and S�U�
5. W�T� 6. T�S� and S�P�
10
8
x � 2
x � 1
SP
M N
L
T
9
103
x � 1
5
P
S
RU V
WT
Q
x � 5
x � 5
9
3
44 S
W
XY
T
U26
14
12 13
B C
DA
13
76 x
G
E H
F
7.5
7.5
7.57.5
Z Y
XW
3
3
33S R
QP
10.5
6 9
59� 35�A C
B
7
4 659� 35�D F
E
© Glencoe/McGraw-Hill 304 Glencoe Geometry
Determine whether each pair of figures is similar. Justify your answer.
1. 2.
Each pair of polygons is similar. Write a similarity statement, and find x, themeasure(s) of the indicated side(s), and the scale factor.
3. L�M� and M�N� 4. D�E� and D�F�
5. COORDINATE GEOMETRY Triangle ABC has vertices A(0, 0), B(�4, 0), and C(�2, 4).The coordinates of each vertex are multiplied by 3 to create �AEF. Show that �AEF issimilar to �ABC.
6. INTERIOR DESIGN Graham used the scale drawing of his living room to decide where to place furniture. Find the dimensions of theliving room if the scale in the drawing is 1 inch � 4.5 feet.
4 in.
21–2 in.
6 12
40�
40�
x � 1
x � 3AF
E
D
B C14
10
x � 6
x � 9
C N MD
A B LP
2412
14 162118
VUAC
B T
25 12
9
14.4
1524
20
15
JM Q R
SP
KL
Practice Similar Polygons
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
Reading to Learn MathematicsSimilar Polygons
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
© Glencoe/McGraw-Hill 305 Glencoe Geometry
Less
on
6-2
Pre-Activity How do artists use geometric patterns?
Read the introduction to Lesson 6-2 at the top of page 289 in your textbook.• Describe the figures that have similar shapes.
• What happens to the figures as your eyes move from the center to theouter edge?
Reading the Lesson1. Complete each sentence.
a. Two polygons that have exactly the same shape, but not necessarily the same size,are .
b. Two polygons are congruent if they have exactly the same shape and the same .
c. Two polygons are similar if their corresponding angles are and their corresponding sides are .
d. Two polygons are congruent if their corresponding angles are and their corresponding sides are .
e. The ratio of the lengths of corresponding sides of two similar figures is called the .
f. Multiplying the coordinates of all points of a figure in the coordinate plane by a scale factor to get a similar figure is called a .
g. If two polygons are similar with a scale factor of 1, then the polygons are .
2. Determine whether each statement is always, sometimes, or never true.a. Two similar triangles are congruent.b. Two equilateral triangles are congruent.c. An equilateral triangle is similar to a scalene triangle.d. Two rectangles are similar.e. Two isosceles right triangles are congruent.f. Two isosceles right triangles are similar.g. A square is similar to an equilateral triangle.h. Two acute triangles are similar.i. Two rectangles in which the length is twice the width are similar.j. Two congruent polygons are similar.
Helping You Remember3. A good way to remember a new mathematical vocabulary term is to relate it to words
used in everyday life. The word scale has many meanings in English. Give three phrasesthat include the word scale in a way that is related to proportions.
© Glencoe/McGraw-Hill 306 Glencoe Geometry
Constructing Similar PolygonsHere are four steps for constructing a polygon that is similar to and withsides twice as long as those of an existing polygon.
Step 1 Choose any point either inside or outside the polygon and label it O.Step 2 Draw rays from O through each vertex of the polygon.Step 3 For vertex V, set the compass to length OV. Then locate a new point
V� on ray OV such that VV� � OV. Thus, OV� � 2(OV ).Step 4 Repeat Step 3 for each vertex. Connect points V�, W�, X� and Y� to
form the new polygon.
Two constructions of polygons similar to and with sides twice those of VWXYare shown below. Notice that the placement of point O does not affect the sizeor shape of V�W�X�Y�, only its location.
Trace each polygon. Then construct a similar polygon with sides twice as long asthose of the given polygon.
1. 2. D E F
GC
A
B
VW
Y X
VW
Y X
V�
W�
X�Y�
O
VW
YX
V�
W�
X�Y�
O
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-26-2
3. Explain how to construct a similarpolygon with sides three times the lengthof those of polygon HIJKL. Then do theconstruction.
4. Explain how to construct a similar
polygon 1�12� times the length of those of
polygon MNPQRS. Then do theconstruction.
M
S
R
N
Q
P
H I
LK
J
Study Guide and InterventionSimilar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
© Glencoe/McGraw-Hill 307 Glencoe Geometry
Less
on
6-3
Identify Similar Triangles Here are three ways to show that two triangles are similar.
AA Similarity Two angles of one triangle are congruent to two angles of another triangle.
SSS Similarity The measures of the corresponding sides of two triangles are proportional.
SAS SimilarityThe measures of two sides of one triangle are proportional to the measures of two corresponding sides of another triangle, and the included angles are congruent.
Determine whether thetriangles are similar.
�DAC
F� � �69� � �
23�
�BE
CF� � �1
82� � �
23�
�DAB
E� � �1105� � �
23�
�ABC � �DEF by SSS Similarity.
15
129
D E
F
10
86
A B
C
Determine whether thetriangles are similar.
�34� � �
68�, so �MQR
N� � �
NRS
P�.
m�N � m�R, so �N � �R.�NMP � �RQS by SAS Similarity.
8
4
70�
Q
R S6
370�
M
N P
Example 1Example 1 Example 2Example 2
ExercisesExercises
Determine whether each pair of triangles is similar. Justify your answer.
1. 2.
3. 4.
5. 6. 32
20
154025
24
3926
1624
18
8
65�9
465�
36
2018
24
© Glencoe/McGraw-Hill 308 Glencoe Geometry
Use Similar Triangles Similar triangles can be used to find measurements.
Study Guide and Intervention (continued)
Similar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
�ABC � �DEF.Find x and y.
�DAC
F� � �BE
CF� �D
ABE� � �
BE
CF�
� �198� �1
y8� � �
198�
18x � 9(18�3�) 9y � 324x � 9�3� y � 36
18�3��x
18��3
B
CA
18 18 9y
x
E
FD
A person 6 feet tall castsa 1.5-foot-long shadow at the same timethat a flagpole casts a 7-foot-longshadow. How tall is the flagpole?
The sun’s rays form similar triangles.Using x for the height of the pole, �
6x� � �
17.5�,
so 1.5x � 42 and x � 28.The flagpole is 28 feet tall.
7 ft
6 ft?
1.5 ft
Example 1Example 1 Example 2Example 2
ExercisesExercises
Each pair of triangles is similar. Find x and y.
1. 2.
3. 4.
5. 6.
7. The heights of two vertical posts are 2 meters and 0.45 meter. When the shorter postcasts a shadow that is 0.85 meter long, what is the length of the longer post’s shadow to the nearest hundredth?
y
x32
10
2230yx
9
8
7.2
16
yx
6038.6
23
3024
36��2yx
3636
20
13
26y
x � 235
13
10 20y
x
Skills PracticeSimilar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
© Glencoe/McGraw-Hill 309 Glencoe Geometry
Less
on
6-3
Determine whether each pair of triangles is similar. Justify your answer.
1. 2.
3. 4.
ALGEBRA Identify the similar triangles, and find x and the measures of theindicated sides.
5. A�C� and E�D� 6. J�L� and L�M�
7. E�H� and E�F� 8. U�T� and R�T�
6
14
S
U
V
R T
x � 4
x � 612
9
6
9
D
E
FH
Gx � 5
16
4M
NL
J
K
x � 18
x � 3
1512
D
E CB
Ax � 1
x � 5
30�60�RQ
M
S
K
T
2170�
15
MJ
P
1470�10
U
S
T
12
128
9
9 6
C P
QR
A
B
2014
13 1221
9
S W X
Y
R
T
© Glencoe/McGraw-Hill 310 Glencoe Geometry
Determine whether each pair of triangles is similar. Justify your answer.
1. 2.
ALGEBRA Identify the similar triangles, and find x and the measures of theindicated sides.
3. L�M� and Q�P� 4. N�L� and M�L�
Use the given information to find each measure.
5. If T�S� || Q�R�, TS � 6, PS � x � 7, 6. If E�F� || H�I�, EF � 3, EG � x � 1,QR � 8, and SR � x � 1, HI � 4, and HG � x � 3,find PS and PR. find EG and HG.
INDIRECT MEASUREMENT For Exercises 7 and 8, use the following information.A lighthouse casts a 128-foot shadow. A nearby lamppost that measures 5 feet 3 inches castsan 8-foot shadow.
7. Write a proportion that can be used to determine the height of the lighthouse.
8. What is the height of the lighthouse?
F
GE
H
I
S RP
QT
8
N
J LK
Mx � 5
6x � 2
24
12
18N
P
QL
M
x � 3x � 1
18
16 12.516
14
11
M
L N
S R
T42�
42�
24
1812
16J
A K
Y S
W
Practice Similar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
Reading to Learn MathematicsSimilar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
© Glencoe/McGraw-Hill 311 Glencoe Geometry
Less
on
6-3
Pre-Activity How do engineers use geometry?
Read the introduction to Lesson 6-3 at the top of page 298 in your textbook.• What does it mean to say that triangular shapes result in rigid
construction?
• What would happen if the shapes used in the construction werequadrilaterals?
Reading the Lesson1. State whether each condition guarantees that two triangles are congruent or similar.
If the condition guarantees that the triangles are both similar and congruent, writecongruent. If there is not enough information to guarantee that the triangles will becongruent or similar, write neither.a. Two sides and the included angle of one triangle are congruent to two sides and the
included angle of the other triangle.b. The measures of all three pairs of corresponding sides are proportional.c. Two angles of one triangle are congruent to two angles of the other triangle.d. Two angles and a nonincluded side of one triangle are congruent to two angles and
the corresponding nonincluded side of the other triangle.e. The measures of two sides of a triangle are proportional to the measures of two
corresponding sides of another triangle, and the included angles are congruent.
f. The three sides of one triangle are congruent to the three sides of the other triangle.
g. The three angles of one triangle are congruent to the three angles of the other triangle.
h. One acute angle of a right triangle is congruent to one acute angle of another righttriangle.
i. The measures of two sides of a triangle are proportional to the measures of two sidesof another triangle.
2. Identify each of the following as an example of a reflexive, symmetric, or transitive property.a. If �RST � �UVW, then �UVW � �RST.b. If �RST � �UVW and �UVW � �OPQ, then �RST � �OPQ.c. �RST � �RST
Helping You Remember3. A good way to remember something is to explain it to someone else. Suppose one of your
classmates is having trouble understanding the difference between SAS for congruenttriangles and SAS for similar triangles. How can you explain the difference to him?
© Glencoe/McGraw-Hill 312 Glencoe Geometry
Ratio Puzzles with TrianglesIf you know the perimeter of a triangle and the ratios of the sides, you canfind the lengths of the sides.
The perimeter of a triangle is 84 units. The sides havelengths r, s, and t. The ratio of s to r is 5:3, and the ratio of t to r is2:1. Find the length of each side.
Since both ratios contain r, rewrite one or both ratios to make r the same. Youcan write the ratio of t to r as 6:3. Now you can write a three-part ratio.r : s : t � 3: 5: 6
There is a number x such that r � 3x, s � 5x, and t � 6x. Since you know theperimeter, 84, you can use algebra to find the lengths of the sides.
r � s � t � 843x � 5x � 6x � 84
14x � 84x � 6
3x � 18, 5x � 30, 6x � 36
So r � 18, s � 30, and t � 36.
Find the lengths of the sides of each triangle.
1. The perimeter of a triangle is 75 units. The sides have lengths a, b, and c. The ratio of b to a is 3:5, and the ratio of c to a is 7:5. Find the length of each side.
2. The perimeter of a triangle is 88 units. The sides have lengths d, e, and f. The ratio of e to d is 3:1, and the ratio of f to e is 10:9. Find the length of each side.
3. The perimeter of a triangle is 91 units. The sides have lengths p, q, and r. The ratio of p to r is 3:1, and the ratio of q to r is 5:2. Find the length of each side.
4. The perimeter of a triangle is 68 units. The sides have lengths g, h, and j. The ratio of j to g is 2:1, and the ratio of h to g is 5:4. Find the length of each side.
5. Write a problem similar to those above involving ratios in triangles.
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-36-3
ExampleExample
Study Guide and InterventionParallel Lines and Proportional Parts
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
© Glencoe/McGraw-Hill 313 Glencoe Geometry
Less
on
6-4
Proportional Parts of Triangles In any triangle, a line parallel to one side of a triangle separates the other two sides proportionally. The converse is also true.
If X and Y are the midpoints of R�T� and S�T�, then X�Y� is amidsegment of the triangle. The Triangle Midsegment Theoremstates that a midsegment is parallel to the third side and is half its length.
If XY��� || RS���, then �RXTX� � �Y
SYT�.
If �RXTX� � �Y
SYT�, then XY��� || RS���.
If X�Y� is a midsegment, then XY��� || RS��� and XY � �12�RS.
Y ST
R
X
In �ABC, E�F� || C�B�. Find x.
Since E�F� || C�B�, �FA
BF� � �E
AEC�.
�xx
��
222
� � �168�
6x � 132 � 18x � 3696 � 12x8 � x
F
C
BA
18
x � 22 x � 2
6E
A triangle has verticesD(3, 6), E(�3, �2), and F(7, �2).Midsegment G�H� is parallel to E�F�. Find the length of G�H�.G�H� is a midsegment, so its length is one-half that of E�F�. Points E and F have thesame y-coordinate, so EF � 7 � (�3) � 10.The length of midsegment G�H� is 5.
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find x.
1. 2. 3.
4. 5. 6.
7. In Example 2, find the slope of E�F� and show that E�F� || G�H�.
30 10x � 10
x11x � 12
x33
30 10
24x
35
x9
20
x
185
5 x
7
© Glencoe/McGraw-Hill 314 Glencoe Geometry
Divide Segments Proportionally When three or more parallel lines cut two transversals,they separate the transversals into proportional parts. If the ratio of the parts is 1, then the parallellines separate the transversals into congruent parts.
If �1 || �2 || �3, If �4 || �5 || �6 and
then �ab� � �d
c�. �
uv� � 1, then �
wx� � 1.
Refer to lines �1, �2 ,and �3 above. If a � 3, b � 8, and c � 5, find d.
�1 || �2 || �3 so �38� � �d
5�. Then 3d � 40 and d � 13�
13�.
Find x and y.
1. 2.
3. 4.
5. 6.16 32
y � 3y
3 4
x � 4x
5
8 y � 2
y2x � 4
3x � 1 2y � 2
3y
2x � 6
x � 3
12
x � 12 12
3x5x
a
bd
u v
xw
ctn
m
s �1
�4 �5 �6
�2
�3
Study Guide and Intervention (continued)
Parallel Lines and Proportional Parts
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
ExampleExample
ExercisesExercises
Skills PracticeParallel Lines and Proportional Parts
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
© Glencoe/McGraw-Hill 315 Glencoe Geometry
Less
on
6-4
1. If JK � 7, KH � 21, and JL � 6, 2. Find x and TV if RU � 8, US � 14,find LI. TV � x � 1 and VS � 17.5.
Determine whether B�C� || D�E�.
3. AD � 15, DB � 12, AE � 10, and EC � 8
4. BD � 9, BA � 27, and CE is one third of EA
5. AE � 30, AC � 45, and AD is twice DB
COORDINATE GEOMETRY For Exercises 6–8, use the following information.Triangle ABC has vertices A(�5, 2), B(1, 8), and C(4, 2). Point Dis the midpoint of A�B� and E is the midpoint of A�C�.
6. Identify the coordinates of D and E.
7. Show that B�C� is parallel to D�E�.
8. Show that DE � �12�BC.
9. Find x and y. 10. Find x and y.
3–2x � 2
x � 3
2y � 1
3y � 52x � 1 3y � 8
y � 5x � 7
x
y
O
A C
B
D
E
C
B
E
D
A
R
U V
T
S
HL
K J
I
© Glencoe/McGraw-Hill 316 Glencoe Geometry
1. If AD � 24, DB � 27, and EB � 18, 2. Find x, QT, and TR if QT � x � 6,find CE. SR � 12, PS � 27, and TR � x � 4.
Determine whether J�K� || N�M�.
3. JN � 18, JL � 30, KM � 21, and ML � 35
4. KM � 24, KL � 44, and NL � �56�JN
COORDINATE GEOMETRY For Exercises 5 and 6, use the following information.Triangle EFG has vertices E(�4, �1), F(2, 5), and G(2, �1). Point Kis the midpoint of E�G� and H is the midpoint of F�G�.
5. Show that E�F� is parallel to K�H�.
6. Show that KH � �12�EF.
7. Find x and y. 8. Find x and y.
9. MAPS The distance from Wilmington to Ash Grove along Kendall is 820 feet and along Magnolia, 660 feet. If the distance between Beech and Ash Grove along Magnolia is 280 feet, what is the distance between the two streets alongKendall?
Ash Grove
Beech
Magnolia Kendall
Wilmington
4–3y � 2
3x � 4 x � 1
4y � 43x � 4
2–3y � 3
1–3y � 65–
4x � 3
x
y
O
E G
F
H
K
N LJ
KM
S
Q
TRP
D
E
C
BA
Practice Parallel Lines and Proportional Parts
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
Reading to Learn MathematicsParallel Lines and Proportional Parts
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
© Glencoe/McGraw-Hill 317 Glencoe Geometry
Less
on
6-4
Pre-Activity How do city planners use geometry?
Read the introduction to Lesson 6-4 at the top of page 307 in your textbook.
Use a geometric idea to explain why the distance between Chicago Avenueand Ontario Street is shorter along Michigan Avenue than along LakeShore Drive.
Reading the Lesson
1. Provide the missing words to complete the statement of each theorem. Then state thename of the theorem.
a. If a line intersects two sides of a triangle and separates the sides into corresponding
segments of lengths, then the line is to the
third side.
b. A midsegment of a triangle is to one side of the triangle and its
length is the length of that side.
c. If a line is to one side of a triangle and intersects the other two sides
in distinct points, then it separates these sides into
of proportional length.
2. Refer to the figure at the right.
a. Name the three midsegments of �RST.
b. If RS � 8, RU � 3, and TW � 5, find the length of each of themidsegments.
c. What is the perimeter of �RST?
d. What is the perimeter of �UVW?
e. What are the perimeters of �RUV, �SVW, and �TUW?
f. How are the perimeters of each of the four small triangles related to the perimeter ofthe large triangle?
g. Would the relationship that you found in part f apply to any triangle in which themidpoints of the three sides are connected?
Helping You Remember
3. A good way to remember a new mathematical term is to relate it to other mathematicalvocabulary that you already know. What is an easy way to remember the definition ofmidsegment using other geometric terms?
R
U
V
W
T
S
© Glencoe/McGraw-Hill 318 Glencoe Geometry
Parallel Lines and Congruent PartsThere is a theorem stating that if three parallel lines cut off congruent segments on onetransversal, then they cut off congruent segments on any transversal. This can be shown forany number of parallel lines. The following drafting technique uses this fact to divide asegment into congruent parts.
A�B� is to be separated into five congruent parts. This can be done very accurately without using a ruler. All that is needed is a compass and a piece of notebook paper.
Step 1 Hold the corner of a piece of notebook paper at point A.
Step 2 From point A, draw a segment along the paper that is five spaces long. Mark where the lines of the notebook paper meet the segment. Label the fifth point, P.
Step 3 Draw P�B�. Through each of the other marks on A�P�, construct a line parallel to B�P�. The points where these lines intersect A�B� will divide A�B� into five congruent segments.
Use a compass and a piece of notebook paper to divide each segment into the given number of congruent parts.
1. six congruent parts
2. seven congruent parts A B
A B
A
P
B
A B
P
A B
A B
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-46-4
Study Guide and InterventionParts of Similar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
© Glencoe/McGraw-Hill 319 Glencoe Geometry
Less
on
6-5
Perimeters If two triangles are similar, their perimeters have the same proportion as the corresponding sides.
If �ABC � �RST, then
�AR
BS
��
BST
C��
RA
TC
� � �AR
BS� � �
BST
C� � �R
ACT�.
Use the diagram above with �ABC � �RST. If AB � 24 and RS � 15,find the ratio of their perimeters.Since �ABC � �RST, the ratio of the perimeters of �ABC and �RST is the same as theratio of corresponding sides.
Therefore � �2145�
� �85�
Each pair of triangles is similar. Find the perimeter of the indicated triangle.
1. �XYZ 2. �BDE
3. �ABC 4. �XYZ
5. �ABC 6. �RST
25
20
10
P T
SM
R
95
13
12
A D
E
C
B
5 6
8.7
13
10
X Y
Z T
R S18
20
25 15 17
8
NB
A C PM
36
9
11C
DE
A
B
44
24
20 22 10
TRX Z
YS
perimeter of �ABC���perimeter of �RST
A C
BS
R T
ExampleExample
ExercisesExercises
© Glencoe/McGraw-Hill 320 Glencoe Geometry
Special Segments of Similar Triangles When two triangles are similar,corresponding altitudes, angle bisectors, and medians are proportional to the correspondingsides. Also, in any triangle an angle bisector separates the opposite side into segments thathave the same ratio as the other two sides of the triangle.
Study Guide and Intervention (continued)
Parts of Similar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
In the figure,�ABC � �XYZ, with angle bisectors asshown. Find x.
Since �ABC � �XYZ, the measures of theangle bisectors are proportional to themeasures of a pair of corresponding sides.
�AX
BY� � �Y
BWD�
�2x4� � �
180�
10x � 24(8)10x � 192
x � 19.2
2410
DW
Y
ZXC
B
A
8x
S�U� bisects �RST. Find x.
Since S�U� is an angle bisector, �RTU
U� � �
RTS
S�.
�2x0� � �
1350�
30x � 20(15)30x � 300
x � 10
x 20
3015
UT
S
R
Example 1Example 1 Example 2Example 2
ExercisesExercises
Find x for each pair of similar triangles.
1. 2.
3. 4.
5. 6. 2x � 2x � 1
281617
11
x � 7
x
10
10x
8
873
3
4x
69
12x
x
36
1820
Skills PracticeParts of Similar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
© Glencoe/McGraw-Hill 321 Glencoe Geometry
Less
on
6-5
Find the perimeter of the given triangle.
1. �JKL, if �JKL � �RST, RS � 14, 2. �DEF, if �DEF � �ABC, AB � 27,ST � 12, TR � 10, and LJ � 14 BC � 16, CA � 25, and FD � 15
3. �PQR, if �PQR � �LMN, LM � 16, 4. �KLM, if �KLM � �FGH, FG � 30,MN � 14, NL � 27, and RP � 18 GH � 38, HF � 38, and KL � 24
Use the given information to find each measure.
5. Find FG if �RST � �EFG, S�H� is an 6. Find MN if �ABC � �MNP, A�D� is an altitude of �RST, F�J� is an altitude of altitude of �ABC, M�Q� is an altitude of�EFG, ST � 6, SH � 5, and FJ � 7. �MNP, AB � 24, AD � 14, and MQ � 10.5.
Find x.
7. �HKL � �XYZ 8.
LH
K
18
10
Y
15
x
ZX7
20
24
x
C
A
BD P NQ
M
JGE
F
HTR
S
GL
K
M
F
H
L MP Q
RN
A
B
C
D
E
F
R S
T
J K
L
© Glencoe/McGraw-Hill 322 Glencoe Geometry
Find the perimeter of the given triangle.
1. �DEF, if �ABC � �DEF, AB � 36, 2. �STU, if �STU � �KLM, KL � 12,BC � 20, CA � 40, and DE � 35 LM � 31, MK � 32, and US � 28
Use the given information to find each measure.
3. Find PR if �JKL � �NPR, K�M� is an 4. Find ZY if �STU � �XYZ, U�A� is an altitude of �JKL, P�T� is an altitude of altitude of �STU, Z�B� is an altitude of �NPR, KL � 28, KM � 18, and �XYZ, UT � 8.5, UA � 6, and PT � 15.75. ZB � 11.4.
Find x.
5. 6.
PHOTOGRAPHY For Exercises 7 and 8, use the following information.Francine has a camera in which the distance from the lens to the film is 24 millimeters.
7. If Francine takes a full-length photograph of her friend from a distance of 3 meters andthe height of her friend is 140 centimeters, what will be the height of the image on thefilm? (Hint: Convert to the same unit of measure.)
8. Suppose the height of the image on the film of her friend is 15 millimeters. If Francinetook a full-length shot, what was the distance between the camera and her friend?
x28
20 30402x � 1 25x � 4
YX B
Z
TS A
U
MJ L
K
TN R
P
KS
LT
UMBE
CFD
A
Practice Parts of Similar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
Reading to Learn MathematicsParts of Similar Triangles
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
© Glencoe/McGraw-Hill 323 Glencoe Geometry
Less
on
6-5
Pre-Activity How is geometry related to photography?
Read the introduction to Lesson 6-5 at the top of page 316 in your textbook.
• How is similarity involved in the process of making a photographic printfrom a negative?
• Why do photographers place their cameras on tripods?
Reading the Lesson
1. In the figure, �RST � �UVW. Complete each proportion involving the lengths of segments inthis figure by replacing the question mark. Thenidentify the definition or theorem from the listbelow that the completed proportion illustrates.
i. Definition of congruent polygons ii. Definition of similar polygons
iii. Proportional Perimeters Theorem iv. Angle Bisectors Theorem
v. Similar triangles have corresponding altitudes proportional to corresponding sides.
vi. Similar triangles have corresponding medians proportional to corresponding sides.
vii. Similar triangles have corresponding angle bisectors proportional to corresponding sides.
a. �RS � S?T � TR� � �U
RSV� b. �U
RWT� � �
S?X�
c. �RU
MN� � �V
?W� d. �U
RSV� � �
S?T�
e. �RPS
P� � �S
?T� f. �
U?N� � �
UR
WT�
g. �WTP
Q� � �R?T� h. �
UVW
W� � �Q
?V�
Helping You Remember
2. A good way to remember a large amount of information is to remember key words. Whatkey words will help you remember the features of similar triangles that are proportionalto the lengths of the corresponding sides?
W
Q N
YU
V
T
P M
XR
S
© Glencoe/McGraw-Hill 324 Glencoe Geometry
Proportions for Similar TrianglesRecall that if a line crosses two sides of a triangle and is parallel to the thirdside, then the line separates the two sides that it crosses into segments ofproportional lengths.
You can write many proportions by identifying similar triangles in thefollowing diagram. In the diagram, AM��� � BN���, AE��� � FL��� � MR���, and DQ��� � ER���.
Answer each question. Use the diagram above.
1. Name a triangle similar to �GNP. 2. Name a triangle similar to �CJH.
3. Name two triangles similar to �JKS. 4. Name a triangle similar to �ACP.
Complete each proportion.
5. �AA
GP� � �
A?F� 6. �C
CHP� � �
C?R� 7. �J
JRS� � �J
?L�
8. �PP
HC� � �
P?G� 9. �
ELR
R� � �J
?R� 10. �
MM
NP� � �A
?P�
Solve.
11. If CJ � 16, JR � 48, and LR � 30, find EL.
12. If DK � 5, KS � 7, and CJ � 8, find JS.
13. If MN � 12, NP � 32, and AP � 48, find AG. Round to the nearest tenth.
14. If CH � 18, HP � 82, and CR � 130, find CJ.
15. Write three more problems that can be solved using the diagram above.
A B C D E
FG H J K L
M N P RQ
S
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-56-5
Study Guide and InterventionFractals and Self-Similarity
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
© Glencoe/McGraw-Hill 325 Glencoe Geometry
Less
on
6-6
Characteristics of Fractals The act of repeating a process over and over, such asfinding a third of a segment, then a third of the new segment, and so on, is called iteration.When the process of iteration is applied to some geometric figures, the results are calledfractals. For objects such as fractals, when a portion of the object has the same shape orcharacteristics as the entire object, the object can be called self-similar.
In the diagram at the right, notice that the details at each stage are similar to the details at Stage 1.
1. Follow the iteration process below to produce a fractal.
Stage 1• Draw a square.• Draw an isosceles right triangle on the top side of the square.
Use the side of the square as the hypotenuse of the triangle.• Draw a square on each leg of the right triangle.
Stage 2Repeat the steps in Stage 1, drawing an isosceles triangle and two small squares for each of the small squares from Stage 1.
Stage 3Repeat the steps in Stage 1 for each of the smallest squares in Stage 2.
2. Is the figure produced in Stage 3 self-similar?
Stage 1 Stage 2 Stage 3
ExercisesExercises
ExampleExample
© Glencoe/McGraw-Hill 326 Glencoe Geometry
Nongeometric Iteration An iterative process can be applied to an algebraicexpression or equation. The result is called a recursive formula.
Find the value of x3, where the initial value of x is 2. Repeat theprocess three times and describe the pattern.
Initial value: 2First time: 23 � 8Second time: 83 � 512Third time: 5123 � 134,217,728
The result of each step of the iteration is used for the next step. For this example, the xvalues are greater with each iteration. There is no maximum value, so the values aredescribed as approaching infinity.
For Exercises 1–5, find the value of each expression. Then use that value as thenext x in the expression. Repeat the process three times, and describe yourobservations.
1. �x�, where x initially equals 5
2. �1x�, where x initially equals 2
3. 3x, where x initially equals 1
4. x � 5 where x initially equals 10
5. x2 � 4, where x initially equals 16
6. Harpesh paid $1000 for a savings certificate. It earns interest at an annual rate of 2.8%,and interest is added to the certificate each year. What will the certificate be worth afterfour years?
Study Guide and Intervention (continued)
Fractals and Self-Similarity
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
ExercisesExercises
ExampleExample
Skills PracticeFractals and Self-Similarity
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
© Glencoe/McGraw-Hill 327 Glencoe Geometry
Less
on
6-6
Stages 1 and 2 of a fractal known as the Cantor set are shown. To get Stage 2, thesegment in Stage 1 is trisected, and the interior of the middle segment is removed.(The interior of a segment is the segment with its endpoints removed.) Theprocess is repeated for subsequent stages.
1. Draw stages 3 and 4 of the Cantor set.
2. How many segments are there in Stage 3? Stage 4?
3. What happens to the length of the line segments in each stage?
4. The Cantor set is the set of points after infinitely many iterations. Is the Cantor set self-similar?
Find the value of each expression. Then, use that value as the next x in theexpression. Repeat the process three times, and describe your observations.
5. 2x � 3, where x initially equals 5
6. �2x
�, where x initially equals 1
Find the first three iterates of each expression.
7. x2 � 1, where x initially equals 2 8. �1x�, where x initially equals 8
Stage 3
Stage 4
Stage 1
Stage 2
© Glencoe/McGraw-Hill 328 Glencoe Geometry
An artist is designing a book cover and wants to show a copy of the book cover inthe lower left corner of the cover. After Stages 1 and 2 of the design, he realizesthat the design is developing into a fractal!
1. Draw Stages 3 and 4 of the book cover fractal.
2. After infinitely many iterations, will the result be a self similar fractal?
3. On what part of the book cover should you focus your attention to be sure you can find acopy of the entire figure?
Find the value of each expression. Then, use that value as the next x in theexpression. Repeat the process three times and describe your observations.
4. 2(x � 3), where x initially equals 0 5. �2x
� � 3, where x initially equals 10
Find the first three iterates of each expression.
6. �2x
� � 1, where x initially equals 1 7. 2x2, where x initially equals 2
8. HOUSING The Andrews purchased a house for $96,000. The real estate agent who soldthe house said that comparable houses in the area appreciate at a rate of 4.5% per year.If this pattern continues, what will be the value of the house in three years? Round tothe nearest whole number.
Stage 3
MATHMATH
MATH
Stage 4
MATHMATH
MATHMATH
Stage 1
MATHStage 2
MATHMATH
Practice Fractals and Self-Similarity
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
Reading to Learn MathematicsFractals and Self-Similarity
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
© Glencoe/McGraw-Hill 329 Glencoe Geometry
Less
on
6-6
Pre-Activity How is mathematics found in nature?
Read the introduction to Lesson 6-6 at the top of page 325 in your textbook.
Name two objects from nature other than broccoli in which a small pieceresembles the whole.
Reading the Lesson1. Match each definition from the first column with a term from the second column. (Some
words of phrases in the second column may be used more than once or not at all.)
Phrase Terma. a geometric figure that is created using iteration i. similar b. a pattern in which smaller and smaller details of a shape have ii. iteration
the same geometric characteristics as the original shape iii. fractalc. the result of translating an iterative process into a formula or iv. self-similar
algebraic equation v. recursiond. the process of repeating the same procedure over and over formula
again vi. congruent
2. Refer to the three patterns below.
i.
ii.
iii.
a. Give the special name for each of the fractals you obtain by continuing the patternwithout end.
b. Which of these fractals are self-similar?
c. Which of these fractals are strictly self-similar?
Helping You Remember3. A good way to remember a new mathematical term is to relate it to everyday English
words. The word fractal is related to fraction and fragment. Use your dictionary to findat least one definition for each of these words that you think is related to the meaning ofthe word fractal and explain the connection.
© Glencoe/McGraw-Hill 330 Glencoe Geometry
The Möbius StripA Möbius strip is a special surface with only one side. It was discovered byAugust Ferdinand Möbius, a German astronomer and mathematician.
1. To make a Möbius strip, cut a strip of paper about 16 inches long and 1inch wide. Mark the ends with the letters A, B, C, and D as shown below.
Twist the paper once, connecting A to D and B to C. Tape the endstogether on both sides.
2. Use a crayon or pencil to shade one side of the paper. Shade around thestrip until you get back to where you started. What happens?
3. What do you think will happen if you cut the Möbius strip down themiddle? Try it.
4. Make another Möbius strip. Starting a third of the way in from one edge,cut around the strip, staying always the same distance in from the edge.What happens?
5. Start with another long strip of paper. Twist the paper twice and connectthe ends. What happens when you cut down the center of this strip?
6. Start with another long strip of paper. Twist the paper three times andconnect the ends. What happens when you cut down the center of this strip?
A
B
C
D
Enrichment
NAME ______________________________________________ DATE ____________ PERIOD _____
6-66-6
Chapter 6 Test, Form 166
© Glencoe/McGraw-Hill 331 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
1. There are 15 plums and 9 apples in a fruit bowl. Find the ratio of apples toplums.A. 3:5 B. 3:8 C. 5:3 D. 8:3
2. The scale drawing of a porch is 8 inches wide by 12 inches long. If the actualporch is 12 feet wide, find the length of the porch.A. 8 ft B. 10 ft C. 16 ft D. 18 ft
3. Solve �56� � �
4x�.
A. 4.6 B. 4.8 C. 5 D. 7
4. A quality control technician checked a sample of 30 bulbs. Two of the bulbswere defective. If the sample was representative, find the number of bulbsexpected to be defective in a case of 450.A. 24 B. 30 C. 36 D. 45
5. Find the triangle similar to �ABC at the right.A. B.
C. D.
6. Find x if �ABC � �JKL.A. 10 B. 14C. 25 D. 29
7. Quadrilateral ABCD � quadrilateral PQRS. If AB � 10, BC � 6, PS � 12,and QR � 4, find the scale factor of ABCD to PQRS.
A. �12� B. �
32� C. �
53� D. �
56�
8. If quadrilateral ABCD � quadrilateralEFGH, find x.A. 15 B. 20C. 25 D. 30
9. Which theorem or postulate can be used to prove that these two triangles are similar?A. AA B. SAS C. SSA D. SSS
10. Find MN.
A. 5�13� B. 6 �
34� C. 7 D. 12
35�8
96
35�
Q
PM
L
N
6
84
27�
63�
B
C
D
A
F
E
H
G 60� 120�80�
100�4x �
A C
B J L
K186 9
x � 2
4
352210
24
15
36
39
5
12
35
12
13
CB
A
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 332 Glencoe Geometry
Chapter 6 Test, Form 1 (continued)66
11.
12.
13.
14.
15.
16.
17.
18.
19.
11. A 5-foot tall student cast a 4-foot shadow. If the tree next to her cast a 44-footshadow, what is the height of the tree?
A. 35�15� ft B. 45 ft C. 51�
12� ft D. 55 ft
12. In �ABC, D�E� || A�C�. If AD � 12, BD � 3, and CE � 10,find BE.A. 1 B. 1 �
12�
C. 2 D. 2 �12�
13. Find x so that A�C� || M�N� in �ABC.A. 8 B. 10C. 25 D. 29
14. Find x.A. 14 B. 15C. 16 D. 18
15. If �FGH � �PQR, FG � 6, PQ � 10, and the perimeter of �PQR is 35, findthe perimeter of �FGH.A. 21 B. 27 C. 31 D. 58�
13�
16. �LMN � �XYZ with altitudes K�L�and W�X�. Find KL.A. 6 B. 7C. 9 D. 19
17. Find x.A. 5 B. 6
C. 6�12� D. 7�
12�
18. Find x.A. 16 B. 18C. 20 D. 21
19. Each arrangement of dots forms a box number. Find the number of dots in the eighth box number.A. 32 B. 48C. 64 D. 512
Bonus In �ABC, AB � 10, BC � 16, D�E� ⊥ A�C�,and DE � 6. Find CD.
106
B C
D
E
A
16
4 8 12
15
1824
x
1610
129x
28 123
K
L X
M YWZN
1610
24x
C B
A
N
M12
18
163x
C B
A
E
D12
10
3
B:
NAME DATE PERIOD
Chapter 6 Test, Form 2A66
© Glencoe/McGraw-Hill 333 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.1. Of the 240 students eating lunch, 96 purchased their lunch and the rest
brought a bag lunch. Find the ratio of students purchasing lunch to studentsbringing a bag lunch.A. 2:3 B. 2:5 C. 3:2 D. 5:2
2. In a rectangle, the ratio of the width to the length is 4:5. If the rectangle is 40 centimeters long, find its width.A. 32 cm B. 36 cm C. 44 cm D. 50 cm
3. A postage stamp 25 millimeters wide and 40 millimeter tall is enlarged tomake a poster. The poster is 4 feet wide. Find the height of the poster.A. 2.5 ft B. 5.25 ft C. 5.8 ft D. 6.4 ft
4. Find the polygon that is similar to ABCD.A. B.
C. D.
5. If �PQR � �STU, find x.A. 4.4 B. 7C. 24.6 D. 35
6. If ABCD � EFGH, find x.A. 18.75 B. 20C. 22.75 D. 28
7. If �ABC � �LMN, AB � 18, BC � 12, LN � 9, and LM � 6, find the scalefactor of �ABC to �LMN.
A. �92� B. �
32� C. �
31� D. �
21�
8. Name the theorem or postulate that can be used to prove that these triangles are similar.A. AA B. SSSC. SAS D. SSA
For Questions 9 and 10, refer to the figure at the right.9. Identify the true statement.
A. �PQR � �RST B. �PQR � �STRC. �PQR � �TSR D. �PQR � �TRS
10. Find x.
A. 2�12� B. 3 C. 3�
12� D. 4
15
18
4x2
21–2
3
T
R
P
Q
S
14
43
7
16
30
10
x � 4
A
BC
DE H
GF
22�123�
5x�12 8
4.5
R U
S
TQ
P
882 6
2–3
218
6
4
12
A D
B C
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 334 Glencoe Geometry
Chapter 6 Test, Form 2A (continued)66
11.
12.
13.
14.
15.
16.
17.
18.
19.
11. A 24-foot flagpole cast a 20-foot shadow. The building next to it cast an 85-footshadow. Find the height of the building.
A. 70�56� ft B. 89 ft C. 96�
16� ft D. 102 ft
12. Find QT.A. 15 B. 17C. 19 D. 21
13. Find x so that S�T� || P�R�.A. 4 B. 4�
12�
C. 6 D. 6�12�
14. Find y.
A. �43� B. 2
C. �73� D. 3
15. If �KLM � �XYZ, find the perimeter of �XYZ.A. 40 B. 42C. 45 D. 48
16. �ABC � �JKL with altitudes B�X� and K�Y�.Find BX.A. 19.2 B. 21C. 24.6 D. 28
17. Find x.A. 4 B. 5C. 6 D. 8
18. Find y.
A. 2�14� B. 2�
34�
C. 3�12� D. 4�
12�
19. Find the third iterate of the expression 2(3x � 1), where x initially equals 1.A. 24 B. 64 C. 302 D. 474
Bonus Find x.12
4 x
6
8
y
DA C
B
6S
Q
P R
12 8
x � 2 x
32
X CA
B10 6
Y LJ
K
9
X Z
Y4
K M
L7
9
69
2y
1–3y � 2
25
5
34x � 3 QR
P
S
T
24
40
35 RQ
PS
T
B:
NAME DATE PERIOD
Chapter 6 Test, Form 2B66
© Glencoe/McGraw-Hill 335 Glencoe Geometry
Ass
essm
ents
Write the letter for the correct answer in the blank at the right of eachquestion.
1. Given the choice between doing an oral and a written report, 18 of the 28 students chose to do an oral report. Find the ratio of written to oral reports.A. 5:9 B. 9:5 C. 9:14 D. 14:9
2. A model of a lighthouse has diameter 8 inches and height 18 inches. If theactual diameter of the lighthouse is 20 feet, find its actual height.A. 30 ft B. 35 ft C. 45 ft D. 50 ft
3. The three sides of a triangle are in the ratio 2:4:5. If the shortest side of thetriangle is 4 meters long, find the perimeter.A. 17 m B. 22 m C. 32 m D. 40 m
4. Find the polygon that may be similar to ABCD.A. B.
C. D.
5. If �ABC � �DEF, find m�C.A. 54° B. 59°C. 67° D. 69°
6. If quadrilateral JKLM � quadrilateral WXYZ, JK � 15, LM � 10, XY � 6,and WX � 9, find KL.A. 8 B. 10 C. 11 D. 12
7. If �LMN � �RST, LN � 21, MN � 28, and the scale factor of �RST to �LMNis �
43�, find ST.
A. 15�34� B. 21 C. 28 D. 37�
13�
8. Name the theorem or postulate that can be used to prove that these triangles are similar.A. AA B. SASC. SSA D. SSS
For Questions 9 and 10, refer to the figure at the right.9. Identify the similar triangles.
A. �LMN � �MPQ B. �LMN � �QMPC. �LMN � �QPM D. �LMN � �PQM
10. Find LM.A. 16 B. 17 C. 18 D. 20
1510 12
L
N
MP
Q
20
8
11.228
54�67�67�
DECA
FB
30
24
8
22
18
6
17
14
6
12
15
5
12
15
54
A B
D C
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 336 Glencoe Geometry
Chapter 6 Test, Form 2B (continued)66
11.
12.
13.
14.
15.
16.
17.
18.
19.
11. A 6-foot tall fence post cast a 2�12�-foot shadow. A nearby clock tower cast a
35-foot shadow. Find the height of the tower.A. 37�
12� ft B. 71 ft C. 78 ft D. 84 ft
12. Find CE.A. 25 B. 26C. 27 D. 28
13. Find FH so that G�H� || D�E�.A. 4 B. 5C. 6 D. 7
14. Find x.A. 4 B. 6C. 9 D. 12
15. If �ABC � �PQR, find the perimeter of �PQR.A. 12 B. 13�
12�
C. 14�12� D. 16
16. �ABC � �XYZ with altitudes A�P�and X�Q�. Find AP.
A. 11�14� B. 14
C. 15�14� D. 20
17. Find x.A. 7 B. 8C. 9 D. 10
18. Find y.
A. 3�34� B. 4
C. 6 D. 7�12�
19. Find the third iterate of the expression 3(2x � 1), where x initially equals 2.A. 27 B. 63 C. 174 D. 303
Bonus Find CE.6
5
4
39
A
B
C
D
E
6
21–4
16
y
R T
SU
40
16
x � 2
3x
125
Q
X
Y Z27
P
A
B C
6 Q
R
P8
7 3
B
C
A
58
2x
x � 3
9
10
15D
G
FE H
8
20
EB C
DA
35
B:
NAME DATE PERIOD
Chapter 6 Test, Form 2C66
© Glencoe/McGraw-Hill 337 Glencoe Geometry
Ass
essm
ents
1. Of the 300 television sets sold at an electronics store lastmonth, 90 were flat-screen TVs. Find the ratio of flat-screenTVs to other TVs sold last month.
2. Determine whether �ABC � �DEF. Justify your answer.
3. When a 5-foot vertical pole casts a 3-foot, 4-inch shadow, anoak tree casts a 20-foot shadow. Find the height of the tree.
4. If quadrilateral ABCD � quadrilateral WXYZ, AB � 15,BC � 27, and the scale factor of WXYZ to ABCD is �
23�, find XY.
5. The blueprint for a swimming pool is 8 inches by 2�12� inches.
The actual pool is 136 feet long. Find the width of the pool.
6. Find CD.
7. If quadrilateral ABCD � quadrilateral PQRS, find BC.
8. Determine whether �ABC � �DEF. Justify your answer.
9. �ABC � �XYZ, AB � 12, AC � 16, BC� 20, and XZ � 24.Find the perimeter of �XYZ.
For Questions 10 and 11, use the figure.
10. Identify the similar triangles.
11. Find x.
15 2x � 4
x6
P
Q
R S
T
9
8
6
42
5
B
EF
DC
A
28
10 6.5R
S P
Q B
A
C
D
1.8
2.46C EG
FD
22.5
15 1510
A
B
C DE
F
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 338 Glencoe Geometry
Chapter 6 Test, Form 2C (continued)66
12. If �ABC � �PQR and B�M� and Q�N� are medians, find BM.
13. The ratio of the measures of the three sides of a triangle is 3:4:6.If the perimeter is 91, find the measure of the longest side.
14. If �RST � �UVW, find m�W.
15. In �ABC, A�X� bisects �BAC. Find x.
16. Find y so that M�N� || B�C�.
17. �ABC � �LMN, and A�D� and L�P� are altitudes. Find AD.
18. Find x.
19. Find the third iterate of the expression x2 � 2, where xinitially equals 1.
Bonus Find EG.18
10
12
B
F
D EG
C
A
28
818
2x � 2
13.5
3620
8D
A
C B M P
L
N
18
8y4
y
B
CAN
M
4x � 5
2x
8
6
A C
X
B
85�
48�R T
S
U W
V
11
4.43.8
A C P R
Q
B
M N
12.
13.
14.
15.
16.
17.
18.
19.
B:
NAME DATE PERIOD
Chapter 6 Test, Form 2D66
© Glencoe/McGraw-Hill 339 Glencoe Geometry
Ass
essm
ents
1. Of the 112 students in the marching band, 35 were in thedrum section. Find the ratio of drummers to other musiciansin the band.
2. Determine whether quadrilateral ABCD � quadrilateralEFGH. Justify your answer.
3. When a 9-foot tall garden shed cast a 5-foot, 3-inch shadow, ahouse cast a 28-foot shadow. Find the height of the house.
4. �ABC � �FGH, AB � 24, AC � 16, GH � 9, and FH � 12.Find the scale factor of �ABC to �FGH.
5. The model of a suspension bridge is 18 inches long and 2 inchestall. If the length of the actual bridge is 1650 feet, find its height.
6. Find GP.
7. If �JKL � �PQR, find x.
8. Determine whether �ABC � �PQR. Justify your answer.
9. �ABC � �PQR, AB � 18, BC � 20, AC � 22, and QR � 25.Find the perimeter of �PQR.
For Questions 10 and 11, use the figure.
10. Identify the similar triangles.
11. Find MN.20
x
6
Y Z
X
M
N
16
20
22.5
3012
158
P
BC
A
Q
R
35
14
6
x
J L
PQ
R
K
2.73.3
4.5
F H
Q
G
P
A
9
7
14 17
9 8.5
13.512
B
C
D H G
FE
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 340 Glencoe Geometry
Chapter 6 Test, Form 2D (continued)66
12. If �FGH � �LMN and A�F� and B�L� are medians, find BL.
13. The ratio of the measures of the three angles of a triangle is3:4:8. Find the measure of the largest angle.
14. If quadrilateral DEFG � quadrilateral WXYZ, find m�Y.
15. In �PQR, Q�S� bisects �PQR. Find x.
16. Find x so that P�Q� || F�H�.
17. If �FGH � �JKL, find GX.
18. Find y.
19. Find the third iterate of the expression 4(x � 2), where xinitially equals 10.
Bonus Find FG.
2.8 3.2 4.2
8 7
A
B F
GEC
D
42 2y
12 16
X Y
K
F H JL
27 12 21–2
G
63
3x2x � 1
F H
Q
G
P
P Q
R
S3x � 1
32
40
2x � 3
E F Z W
XY
GD
68� 85�
82�125�
16.8 4.8
7.7
F H L
MB
N
G
A
12.
13.
14.
15.
16.
17.
18.
19.
B:
NAME DATE PERIOD
Chapter 6 Test, Form 366
© Glencoe/McGraw-Hill 341 Glencoe Geometry
Ass
essm
ents
1. In an orchard of apple and peach trees, �37� of the trees are
peaches. Find the ratio of apple trees to peach trees.
2. Determine whether trapezoid ABCD � trapezoid PQRS.Justify your answer.
3. In �ABC, m�A � 51, AB � 14, and AC � 20. In �DEF,m�D � 51, DE � 16.8, and DF � 24. Determine whether�ABC � �DEF. Justify your answer.
4. A painting that is 48 inches by 12 inches is reduced to fit onan area that is 30 centimeters by 10 centimeters. Find themaximum dimensions of the reduced painting.
5. Find x.
6. If �ABC � �XYZ, find y.
7. The ratio of the measures of the sides of a triangle is 2:5:6. Ifthe length of the longest side is 48 inches, find the perimeter.
8. Find m�I.
9. �ABC � �DEF, AB � 8, BC � 13, AC � 15, and DF � 20.Find the perimeter of �DEF.
10. �ABC � �JKL, AB � 12, BC � 18.4, KL � 6.9, and JL � 5.6.Find the scale factor of �ABC to �JKL.
11. Find y.y � 8
4y � 7
F
G
HI
J45�
53�
17.5A C
B
2y
14X Z
Yy � 3
6
4.8
6.5K L
J
N
M
2x
9.6
15.6
12.47.8
9
15
6
23.4D
R S
PQ
C
A
B
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 342 Glencoe Geometry
Chapter 6 Test, Form 3 (continued)66
12. Find SR.
For Questions 13 and 14, �FGH � �JKL.
13. Find x.
14. Find the ratio of the perimeter of �FGH to the perimeter of �JKL.
15. Find AD so that D�E� || A�B�.
16. Find the fourth iterate of the expression 3x � 1, where xinitially equals 0.8.
17. When a 15-foot tall climbing wall cast a 20-foot shadow, abuilding cast a 32-foot shadow. Find the height of the building.
18. If �ABC � �EFG and B�D� and F�H� are medians, find BD.
19. Stage 1 is repeated in the upper left half of the unshaded square of each stage. Find the perimeter of thesmallest square in Stage 3.
Bonus The ratio of the lengths of the sides of �ABC is 4:5:2.5. The scale factor of �ABC to �DECis 5:2, and the scale factor of�DEC to �FEG is 1:4. B�C� is the shortest side and A�B� is thelongest side of �ABC. Find FG.
10
A
B C
D E
G
F
Stage 1
10 in.
Stage 2
HE G
F2x � 5
751–4
x � 5
DA C
B
A B
ED
C
2x � 4 3x
8x5x
12x
F H
G7.5
4.5
J L
K
30 12Q
RPS
40
12.
13.
14.
15.
16.
17.
18.
19.
B:
NAME DATE PERIOD
Chapter 6 Open-Ended Assessment66
© Glencoe/McGraw-Hill 343 Glencoe Geometry
Ass
essm
ents
Demonstrate your knowledge by giving a clear, concise solution toeach problem. Be sure to include all relevant drawings and justifyyour answers. You may show your solution in more than one way orinvestigate beyond the requirements of the problem.
1. Write a possible proportion if the extremes are 3 and 10.
2. �ABC and �WXY are isosceles triangles.
a. Write a possible ratio for the sides of �ABC if its perimeter is 42 inches.
b. Name possible measures for the sides of �ABC using your answer toPart a.
c. If �WXY has a perimeter of 28 and �ABC has sides with the measuresyou gave in Part b, what must be the measure of the sides of �WXY sothat �WXY � �ABC?
3. Write as many triangle similarity statements as possible for the figurebelow. How do you know that these triangles are similar?
4. Sketch two triangles that are not similar, but have one pair of correspondingangles congruent and two pairs of corresponding sides proportional. Labelthe corresponding angles and the proportional sides.
5.
a. Draw �XYZ inside �PQR with half the perimeter of �PQR. Explainyour process and why it works.
b. Draw the figure that results when your process in Part a is iterated 3 times.
c. Explain why your figure in Part b is a fractal.
d. The perimeter of �PQR is �12�x. If x equals 96, what is the perimeter of
the smallest triangle in your figure in Part b?
P R
Q
A B C D
I E
F
G
H
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 344 Glencoe Geometry
Chapter 6 Vocabulary Test/Review66
Choose from the terms above to complete each sentence.
1. If there are 15 girls and 9 boys in an art class, the ofgirls to boys in the class is 5:3.
2. If �ABC � �DEF, AB � 10, and DE � 2.5, then the of�ABC to �DEF is 4:1.
3. In �LMN, P lies on L�M� and Q on L�N�. If PQ � �12�MN, P�Q� is
called a(n) .
4. The product of the in the equation �3x� � �
2340� is 90.
5. The product of the in the equation �3x� � �
2340� is 24x.
6. A geometric figure created by using iteration is called a(n).
7. If quadrilaterals ABCD and WXYZ have corresponding anglescongruent and corresponding sides proportional, they arecalled .
8. In a recursive formula, an algebraic expression is evaluatedrepeatedly in a process called .
9. The equation �3x� � �
2340� is called a(n) .
10. If some parts of a figure resemble the figure as a whole, thefigure is called .
In your own words—
11. Explain what is meant by equality of cross products.
12. Describe a self-similar figure.
?
?
?
?
?
?
?
?
?
? 1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
cross productsextremesfractal
iterationmeansmidsegment
proportionratioscale factor
self-similarsimilar polygons
NAME DATE PERIOD
SCORE
Chapter 6 Quiz (Lessons 6–1 and 6–2)
66
© Glencoe/McGraw-Hill 345 Glencoe Geometry
Ass
essm
ents
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
1. GARDENS The model of a circular garden is 8 inches indiameter. The actual garden will be 20 feet in diameter. Whatis the ratio of the diameter of the model to the diameter of theactual garden?
2. PHOTOS A 4-inch by 6-inch photograph, set vertically, isenlarged to make a poster 22 inches wide. How tall is the poster?
3. Are any of the three trianglessimilar? If so, write theappropriate similaritystatement.
4. If �ABC � �DEC, find x and the scale factor of �ABC to �DEC.
5. STANDARDIZED TEST PRACTICE The perimeter of arectangle is 126 centimeters. The ratio of the length to thewidth is 5:2. Find the width of the rectangle.A. 9 cm B. 18 cm C. 45 cm D. 50.4 cm
D
E
CB
A
20
253x � 2
2x
BP X Y
Z
Q
R
A
C
5
4 8 8 8
10
4
Chapter 6 Quiz (Lesson 6–3)
66
1.
2.
3.
4.
5.
For Questions 1 and 2, determine whether each pair oftriangles is similar. Justify your answer.
1. 2.
3. Identify the similar triangles in the figure,then find x.
4. SHADOWS A person who is 5 feet tall casts a shadow that is4 feet long. At the same time, a flagpole casts a shadow that is18 feet long. How tall is the flagpole?
5. Find x.
46
9 x
8
A
B D
EF
C 4
6x
20
15
98
10 1241�
52�41�
87�
NAME DATE PERIOD
SCORE
© Glencoe/McGraw-Hill 346 Glencoe Geometry
Chapter 6 Quiz (Lessons 6–4 and 6–5)
66
1.
2.
3.
4.
5.
1. If �PQR � � XYZ, find the perimeter of �XYZ.
2. Find x so that L�M� || A�B�.
3. In �ABC, D�E� is parallel to A�C� and DE � 10. What is thelength of A�C� if D�E� is the midsegment of �ABC?
4. Find DE.
5. In �RST, TU��� bisects �T. If U is a point on R�S�, RU � 6,RT � 9, and ST � 12, find RS.
20
12
15 DE
B
A
C
6 8
4A B
C
L Mx
16
128
P R
Q
5
X Z
Y
NAME DATE PERIOD
SCORE
Chapter 6 Quiz (Lesson 6–6)
66
1.
2.
3.
4.
1. DIAMOND NUMBERS The dots in each arrangement form a diamondnumber. How many dots are in the ninth diamond number?
2. In the fractal, Stage 1 is repeated in the upper right and lowerleft unshaded squares of each stage. Draw Stage 3.
For Questions 3 and 4, find the first 3 iterates of the givenexpression.
3. 4x � 1, where x initially equals 2
4. x2 � x, where x initially equals 3
Stage 1 Stage 2
Stage 3
NAME DATE PERIOD
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Chapter 6 Mid-Chapter Test (Lessons 6–1 through 6–3)
66
© Glencoe/McGraw-Hill 347 Glencoe Geometry
Ass
essm
ents
1. If quadrilateral ABCD � quadrilateral PQRS, which proportion must be true?
A. �AA
DC� � �
PP
QS� B. �C
BDC� � �
QR
RS� C. �B
ADB� � �Q
PQR� D. �
CA
DB� � �
PR
QS�
2. This fall 126 students participated in the soccer program, while 54 playedvolleyball. What was the ratio of soccer players to volleyball players?
A. �34� B. �
37� C. �
43� D. �
73�
3. The ratio of the measures of the angles of a triangle is 2:3:10. What is theleast angle measure?A. 12 B. 15 C. 24 D. 36
4. Find x.A. 2 B. 4.8C. 6 D. 6.4
5. If rectangle ABCD � rectangle EFGH, the perimeter of ABCD is 54 centimeters, and the perimeter of EFGH is 36 centimeters, what is thescale factor of ABCD to EFGH?
A. �23� B. �
32� C. �
35� D. �
53�
10 8
x
D
E
AC B18
6.
7.
8.
9.
10.
NAME DATE PERIOD
SCORE
1.
2.
3.
4.
5.
Part II
For Questions 6 and 7, determine whether each pair oftriangles is similar. Justify your answer.
6. 7.
8. If �ABE � �BCD, find DE and the scale factor of �ABE to �BCD.
9. Quadrilateral ABCD � quadrilateral RSUV, m�ABC � 120,and the scale factor of ABCD to RSUV is �
85�. What is m�RSU?
10. MODELS Sasha made a model of a clipper ship. If her modelhas a length of 18 inches, and the original ship had a length of160 feet and a width of 32 feet, what should be the width ofher model?
24 32
9
A
B ED
C
45251810 85�85�6
415
10
Part I Write the letter for the correct answer in the blank at the right of each question.
© Glencoe/McGraw-Hill 348 Glencoe Geometry
Chapter 6 Cumulative Review(Chapters 1–6)
66
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
1. Find the coordinates of the midpoint of A�B� for A(�24, 15) andB(13, �31). (Lesson 1-3)
For Questions 2–4, use the figure at the right.2. The perimeter of rectangle DEFG is 176,
EF � h, and DE � 7h. Find h. (Lesson 1-6)
3. If p: DEFG is a rectangle and q: DE � EF, find the truth valueof p � q. (Lesson 2-2)
4. How many line segments can be drawn connecting D, E, F,and G? (Lesson 2-5)
5. For S(�5, 7), T(1, 9), P(12, �1), and R(3, 26), determinewhether ST��� and PR��� are parallel, perpendicular, or neither.(Lesson 3-3)
6. Write an equation in slope-intercept form for the line y � 7 � 4(x �10). (Lesson 3-4)
7. Given T(3, �1), U(1, �7), V(8, �5), W(2, 6), X(�4, 8), andY(�2, 1) determine whether �TUV � �WXY. Explain.(Lesson 4-4)
8. Name a pair of congruent angles and a pair of congruent sides that are not marked in the figure. (Lesson 4-6)
For Questions 9 and 10,use the figure at the right.9. Points D, E, and F are
the midpoints of A�B�, B�C�,and C�A�, respectively.Find x, y, and z. (Lesson 5-1)
10. Compare m�BEG to m�CEG. (Lesson 5-5)
For Questions 11 and 12, use the figure at the right.11. Determine whether �HJL � �HKM.
Justify your answer. (Lesson 6-3)
12. If JK � 7, HJ � 8, HL � 12, and LM � 2x � 7.5, find x.(Lesson 6-4)
13. Find a and b. (Lesson 6-4)
L3a � 2
3b
b � 46 � a
M
N
R
Q
P
H ML
K
J
B
CAF 3y � 1
8 � 5z
2.57
1.5
x � 3D EG
L N
M
P
QR
D E
FG
NAME DATE PERIOD
SCORE
Standardized Test Practice (Chapters 1–6)
© Glencoe/McGraw-Hill 349 Glencoe Geometry
1. Find the length of Y�Z�. (Lesson 1-2)
A. 1.9 in. B. 5.3 in.C. 7.2 in. D. 12.5 in.
2. Given: 3b � 4 � 16 Conjecture: b � 0Which of the following would be a counterexample? (Lesson 2-1)
E. b � �1 F. b � 0 G. b � 3.5 H. b � 4
3. Find the plane that is parallel to plane PTU. (Lesson 3-1)
A. plane QRU B. plane QRSC. plane PQS D. plane SPU
4. Find m�1. (Lesson 3-2)
E. 5 F. 12G. 41 H. 44
5. Which statement must be true in order to prove �ABC � �DCB by SAS? (Lesson 4-4)
A. C�B� bisects �ABD.B. �BCA � �CBDC. �BDC � �CABD. A�B� � B�C�
6. In an indirect proof, you assume that the conclusion is false andthen find a(n) . (Lesson 5-3)
E. assumption F. contradictionG. truth value H. conditional statement
7. Demont and Tony are competing to see whose house is the tallest.Early in the afternoon, Tony, who is 4 feet tall, measured his shadowto be 9.6 inches and the shadow of his house to be 62.4 inches. Laterin the day, Demont, who is 5 feet tall, measured his shadow to be15.6 inches and the shadow of his house to be 62.4 inches. Who livesin the taller house? (Lesson 6-3)
A. Demont B. Both houses are the same height.C. Tony D. There is not enough information.
8. Which geometric figure is created using iteration? (Lesson 6-6)
E. scalene triangle F. squareG. fractal H. midsegment
?
D
A C
B
(3x � 5)�
(125 � 7x)�
1
R
P
Q
STU
X Z5.5 in.
3.6 in. Y
NAME DATE PERIOD
SCORE
Part 1: Multiple Choice
Instructions: Fill in the appropriate oval for the best answer.
1.
2.
3.
4.
5.
6.
7.
8. E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
Ass
essm
ents
66
© Glencoe/McGraw-Hill 350 Glencoe Geometry
Standardized Test Practice (continued)
9. What is the distance between �76 and 43 on anumber line? (Lesson 1-3)
10. If 28 � 4x � �12, then 5 � 2x � .(Lesson 2-6)
11. Find m�2. (Lesson 3-2)
12. �LMN is equilateral, LM is one more thanthree times a number, MN is nine less than fivetimes the number, and NL is eleven more thanthe number. Find LM. (Lesson 4-1)
52�
12 34
?
NAME DATE PERIOD
Part 2: Grid In
Instructions: Enter your answer by writing each digit of the answer in a column boxand then shading in the appropriate oval that corresponds to that entry.
Part 3: Short Response
Instructions: Show your work or explain in words how you found your answer.
9. 10.
11. 12.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
13. Solve ��4106
� � �4x �
510
�. (Lesson 6-1)
14. X�A� is an altitude of �XYB. Find y.(Lesson 5-1)
15. Two sides of a triangle measure 21 inches and 32 inches, andthe third side measures x inches. Find the range for x. (Lesson 5-4)
16. If �ABC � �HIJ, find the perimeter of �HIJ. (Lesson 6-5) 32 cm
80 cm
60 cm
A B
C 12 cmH I
J
(2y � 81)� BA
X
Y
13.
14.
15.
16.
1 1 9 2 5
1 2 8 1 6
66
Standardized Test PracticeStudent Record Sheet (Use with pages 338–339 of the Student Edition.)
66
© Glencoe/McGraw-Hill A1 Glencoe Geometry
An
swer
s
Select the best answer from the choices given and fill in the corresponding oval.
1 4 7
2 5 8
3 6 DCBADCBA
DCBADCBADCBA
DCBADCBADCBA
NAME DATE PERIOD
Part 1 Multiple ChoicePart 1 Multiple Choice
Part 2 Short Response/Grid InPart 2 Short Response/Grid In
Part 3 Open-EndedPart 3 Open-Ended
Solve the problem and write your answer in the blank.
For Questions 11 and 12, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.
9 11 12
10
11 (grid in)
12 (grid in)
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
Record your answers for Questions 13–14 on the back of this paper.
© Glencoe/McGraw-Hill A2 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Pro
po
rtio
ns
NA
ME
____
____
____
____
____
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____
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6-1
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Lesson 6-1
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ties
.Th
e ra
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e b
is n
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,can
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tten
as
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.Th
e ra
tio
of t
wo
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titi
es i
s so
met
imes
cal
led
a sc
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fact
or.F
or a
sca
le f
acto
r,th
e u
nit
s fo
r ea
ch q
uan
tity
are
th
e sa
me.
In 2
002,
the
Ch
icag
o C
ub
s b
aseb
all
team
won
67
gam
es o
ut
of 1
62.
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te a
rat
io f
or t
he
nu
mb
er o
f ga
mes
won
to
the
tota
l n
um
ber
of
gam
es p
laye
d.
To
fin
d th
e ra
tio,
divi
de t
he
nu
mbe
r of
gam
es w
on b
y th
e to
tal
nu
mbe
r of
gam
es p
laye
d.T
he
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lt i
s � 16 67 2�
,wh
ich
is
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41.T
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icag
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abo
ut
41%
of
thei
r ga
mes
in
200
2.
A d
oll
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se t
hat
is
15 i
nch
es t
all
is a
sca
le m
odel
of
a re
al h
ouse
wit
h a
hei
ght
of 2
0 fe
et.W
hat
is
the
rati
o of
th
e h
eigh
t of
th
e d
oll
hou
se t
o th
eh
eigh
t of
th
e re
al h
ouse
?T
o st
art,
con
vert
th
e h
eigh
t of
th
e re
al h
ouse
to
inch
es.
20 f
eet
�12
in
ches
per
foo
t �
240
inch
esT
o fi
nd
the
rati
o or
sca
le f
acto
r of
th
e h
eigh
ts,d
ivid
e th
e h
eigh
t of
th
e do
ll h
ouse
by
the
hei
ght
of t
he
real
hou
se.T
he
rati
o is
15
inch
es:2
40 i
nch
es o
r 1:
16.T
he
hei
ght
of t
he
doll
hou
se i
s � 11 6�
the
hei
ght
of t
he
real
hou
se.
1.In
th
e 20
02 M
ajor
Lea
gue
base
ball
sea
son
,Sam
my
Sos
a h
it 4
9 h
ome
run
s an
d w
as a
tba
t 55
6 ti
mes
.Fin
d th
e ra
tio
of h
ome
run
s to
th
e n
um
ber
of t
imes
he
was
at
bat.
� 54 59 6�
2.T
her
e ar
e 18
2 gi
rls
in t
he
soph
omor
e cl
ass
of 3
05 s
tude
nts
.Fin
d th
e ra
tio
of g
irls
to
tota
l st
ude
nts
.
�1 38 02 5�
3.T
he
len
gth
of
a re
ctan
gle
is 8
in
ches
an
d it
s w
idth
is
5 in
ches
.Fin
d th
e ra
tio
of l
engt
h
to w
idth
.
�8 5�
4.T
he
side
s of
a t
rian
gle
are
3 in
ches
,4 i
nch
es,a
nd
5 in
ches
.Fin
d th
e sc
ale
fact
or b
etw
een
the
lon
gest
an
d th
e sh
orte
st s
ides
.
�5 3�
5.T
he
len
gth
of
a m
odel
tra
in i
s 18
in
ches
.It
is a
sca
le m
odel
of
a tr
ain
th
at i
s 48
fee
t lo
ng.
Fin
d th
e sc
ale
fact
or.
� 31 2�
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill29
6G
lenc
oe G
eom
etry
Use
Pro
per
ties
of
Pro
po
rtio
ns
A s
tate
men
t th
at t
wo
rati
os a
re e
qual
is
call
ed a
pro
por
tion
.In
th
e pr
opor
tion
�a b��
� dc � ,w
her
e b
and
dar
e n
ot z
ero,
the
valu
es a
and
dar
e th
e ex
trem
esan
d th
e va
lues
ban
d c
are
the
mea
ns.
In a
pro
port
ion
,th
e pr
odu
ct o
f th
em
ean
s is
equ
al t
o th
e pr
odu
ct o
f th
e ex
trem
es,s
o ad
�bc
.
�a b��
� dc �
a�
d�
b�
c↑
↑ex
trem
esm
eans
Sol
ve � 19 6�
��2 x7 �
.
� 19 6��
�2 x7 �
9 �
x�
16 �
27C
ross
pro
duct
s
9x�
432
Mul
tiply
.
x�
48D
ivid
e ea
ch s
ide
by 9
.
A r
oom
is
49 c
enti
met
ers
by
28 c
enti
met
ers
on a
sca
le d
raw
ing
of a
hou
se.F
or t
he
actu
al r
oom
,th
e la
rger
dim
ensi
on i
s 14
fee
t.F
ind
th
e sh
orte
rd
imen
sion
of
the
actu
al r
oom
.If
xis
th
e ro
om’s
sh
orte
r di
men
sion
,th
en
�2 48 9��
� 1x 4�
49x
�39
2C
ross
pro
duct
s
x�
8D
ivid
e ea
ch s
ide
by 4
9.
Th
e sh
orte
r si
de o
f th
e ro
om i
s 8
feet
.
Sol
ve e
ach
pro
por
tion
.
1.�1 2�
��2 x8 �
562.
�3 8��
� 2y 4�9
3.�x x� �
2 22�
��3 10 0�
8
4.� 18
3 .2��
�9 y�54
.65.
�2x8�
3�
��5 4�
3.5
6.�x x
� �1 1
��
�3 4��
7
Use
a p
rop
orti
on t
o so
lve
each
pro
ble
m.
7.If
3 c
asse
ttes
cos
t $4
4.85
,fin
d th
e co
st o
f on
e ca
sset
te.
$14.
95
8.T
he
rati
o of
th
e si
des
of a
tri
angl
e ar
e 8:
15:1
7.If
th
e pe
rim
eter
of
the
tria
ngl
e is
48
0 in
ches
,fin
d th
e le
ngt
h o
f ea
ch s
ide
of t
he
tria
ngl
e.96
in.,
180
in.,
204
in.
9.T
he
scal
e on
a m
ap i
ndi
cate
s th
at o
ne
inch
equ
als
4 m
iles
.If
two
tow
ns
are
3.5
inch
esap
art
on t
he
map
,wh
at i
s th
e ac
tual
dis
tan
ce b
etw
een
th
e to
wn
s?14
mi
shor
ter
dim
ensi
on�
��
long
er d
imen
sion
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Pro
po
rtio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-1
6-1
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Answers (Lesson 6-1)
© Glencoe/McGraw-Hill A3 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Pro
po
rtio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-1
6-1
©G
lenc
oe/M
cGra
w-H
ill29
7G
lenc
oe G
eom
etry
Lesson 6-1
1.FO
OTB
ALL
A t
igh
t en
d sc
ored
6 t
ouch
dow
ns
in 1
4 ga
mes
.Fin
d th
e ra
tio
of t
ouch
dow
ns
per
gam
e.
0.43
:1
2.E
DU
CA
TIO
NIn
a s
ched
ule
of
6 cl
asse
s,M
arta
has
2 e
lect
ive
clas
ses.
Wh
at i
s th
e ra
tio
of e
lect
ive
to n
on-e
lect
ive
clas
ses
in M
arta
’s s
ched
ule
?
1:2
3.B
IOLO
GY
Ou
t of
274
lis
ted
spec
ies
of b
irds
in
th
e U
nit
ed S
tate
s,78
spe
cies
mad
e th
een
dan
gere
d li
st.F
ind
the
rati
o of
en
dan
gere
d sp
ecie
s of
bir
ds t
o li
sted
spe
cies
in
th
eU
nit
ed S
tate
s.
� 13 39 7�
4.A
RT
An
art
ist
in P
ortl
and,
Ore
gon
,mak
es b
ron
ze s
culp
ture
s of
dog
s.T
he
rati
o of
th
eh
eigh
t of
a s
culp
ture
to
the
actu
al h
eigh
t of
th
e do
g is
2:3
.If
the
hei
ght
of t
he
scu
lptu
reis
14
inch
es,f
ind
the
hei
ght
of t
he
dog.
21 in
.
5.SC
HO
OL
Th
e ra
tio
of m
ale
stu
den
ts t
o fe
mal
e st
ude
nts
in
th
e dr
ama
clu
b at
Cam
pbel
lH
igh
Sch
ool
is 3
:4.I
f th
e n
um
ber
of m
ale
stu
den
ts i
n t
he
clu
b is
18,
wh
at i
s th
e n
um
ber
of f
emal
e st
ude
nts
?
24
Sol
ve e
ach
pro
por
tion
.
6.�2 5�
�� 4x 0�
167.
� 17 0��
�2 x1 �30
8.�2 50 �
��4 6x �
6
9.�5 4x �
��3 85 �
3.5
10.�
x� 3
1�
��7 2�
9.5
11.�
1 35 ��
�x� 5
3�
28
Fin
d t
he
mea
sure
s of
th
e si
des
of
each
tri
angl
e.
12.T
he
rati
o of
th
e m
easu
res
of t
he
side
s of
a t
rian
gle
is 3
:5:7
,an
d it
s pe
rim
eter
is
450
cen
tim
eter
s.90
cm
,150
cm
,210
cm
13.T
he r
atio
of
the
mea
sure
s of
the
sid
es o
f a
tria
ngle
is 5
:6:9
,and
its
peri
met
er is
220
met
ers.
55 m
,66
m,9
9 m
14.T
he
rati
o of
th
e m
easu
res
of t
he
side
s of
a t
rian
gle
is 4
:6:8
,an
d it
s pe
rim
eter
is
126
feet
.28
ft,
42 f
t,56
ft
15.T
he r
atio
of
the
mea
sure
s of
the
sid
es o
f a
tria
ngle
is
5:7:
8,an
d it
s pe
rim
eter
is
40 i
nche
s.10
in.,
14 in
.,16
in.
©G
lenc
oe/M
cGra
w-H
ill29
8G
lenc
oe G
eom
etry
1.N
UTR
ITIO
NO
ne o
unce
of
ched
dar
chee
se c
onta
ins
9 gr
ams
of f
at.S
ix o
f th
e gr
ams
of f
atar
e sa
tura
ted
fats
.Fin
d th
e ra
tio
of s
atur
ated
fat
s to
tot
al f
at i
n an
oun
ce o
f ch
eese
.
2:3
2.FA
RM
ING
Th
e ra
tio
of g
oats
to
shee
p at
a u
niv
ersi
ty r
esea
rch
far
m i
s 4:
7.T
he
nu
mbe
rof
sh
eep
at t
he
farm
is
28.W
hat
is
the
nu
mbe
r of
goa
ts?
16
3.A
RT
Edw
ard
Hop
per’
s oi
l on
can
vas
pain
tin
g N
igh
thaw
ksh
as a
len
gth
of
60 i
nch
es a
nd
a w
idth
of
30 i
nch
es.A
pri
nt
of t
he
orig
inal
has
a l
engt
h o
f 2.
5 in
ches
.Wh
at i
s th
e w
idth
of t
he
prin
t?
1.25
in.
Sol
ve e
ach
pro
por
tion
.
4.�5 8�
�� 1x 2�
7.5
5.� 1.
x 12��
�1 5�0.
224
6.� 26 7x �
��4 3�
6
7.�x
� 32
��
�8 9��2 3�
8.�3x
4�5
��
�� 75 ��5 7�
9.�x
� 42
��
�x� 2
4�
�10
Fin
d t
he
mea
sure
s of
th
e si
des
of
each
tri
angl
e.
10.T
he
rati
o of
th
e m
easu
res
of t
he
side
s of
a t
rian
gle
is 3
:4:6
,an
d it
s pe
rim
eter
is
104
feet
.
24 f
t,32
ft,
48 f
t
11.T
he r
atio
of
the
mea
sure
s of
the
sid
es o
f a
tria
ngle
is 7
:9:1
2,an
d it
s pe
rim
eter
is 8
4 in
ches
.
21 in
.,27
in.,
36 in
.
12.T
he
rati
o of
th
e m
easu
res
of t
he
side
s of
a t
rian
gle
is 6
:7:9
,an
d it
s pe
rim
eter
is
77 c
enti
met
ers.
21 c
m,2
4.5
cm,3
1.5
cm
Fin
d t
he
mea
sure
s of
th
e an
gles
in
eac
h t
rian
gle.
13.T
he
rati
o of
th
e m
easu
res
of t
he
angl
es i
s 4:
5:6.
48,6
0,72
14.T
he
rati
o of
th
e m
easu
res
of t
he
angl
es i
s 5:
7:8.
45,6
3,72
15.B
RID
GES
Th
e sp
an o
f th
e B
enja
min
Fra
nkl
in s
usp
ensi
on b
ridg
e in
Ph
ilad
elph
ia,
Pen
nsy
lvan
ia,i
s 17
50 f
eet.
A m
odel
of
the
brid
ge h
as a
spa
n o
f 42
in
ches
.Wh
at i
s th
era
tio
of t
he
span
of
the
mod
el t
o th
e sp
an o
f th
e ac
tual
Ben
jam
in F
ran
klin
Bri
dge?
� 51 00�
Pra
ctic
e (
Ave
rag
e)
Pro
po
rtio
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-1
6-1
Answers (Lesson 6-1)
© Glencoe/McGraw-Hill A4 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csP
rop
ort
ion
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-1
6-1
©G
lenc
oe/M
cGra
w-H
ill29
9G
lenc
oe G
eom
etry
Lesson 6-1
Pre-
Act
ivit
yH
ow d
o ar
tist
s u
se r
atio
s?
Rea
d th
e in
trod
uct
ion
to
Les
son
6-1
at
the
top
of p
age
282
in y
our
text
book
.
Est
imat
e th
e ra
tio
of l
engt
h t
o w
idth
for
th
e ba
ckgr
oun
d re
ctan
gles
in
Tif
fan
y’s
Cle
mat
is S
kyli
ght.
Sam
ple
an
swer
:ab
ou
t 5
to 2
Rea
din
g t
he
Less
on
1.M
atch
eac
h d
escr
ipti
on i
n t
he
firs
t co
lum
n w
ith
a w
ord
or p
hra
se f
rom
th
e se
con
dco
lum
n.
a.T
he
rati
o of
tw
o co
rres
pon
din
g qu
anti
ties
ivi.
prop
orti
on
b.
ran
d u
in t
he
equ
atio
n �r s�
�� ut �
vii
.cro
ss p
rodu
cts
c.a
com
pari
son
of
two
quan
titi
esvi
iii.
mea
ns
d.
ruan
d st
in t
he
equ
atio
n �r s�
�� ut �
iiiv
.sc
ale
fact
or
e.an
equ
atio
n s
tati
ng
that
tw
o ra
tios
are
equ
ali
v.ex
trem
es
f.s
and
tin
th
e eq
uat
ion
�r s��
� ut �iii
vi.r
atio
2.If
m,n
,p,a
nd
qar
e n
onze
ro n
um
bers
su
ch t
hat
�m n��
�p q� ,w
hic
h o
f th
e fo
llow
ing
stat
emen
ts c
ould
be
fals
e?B
,C,E
A.
np
�m
qB
.�np �
�� mq �
C.
mp
�n
qD
.qm
�pn
E.
�m n��
� pq �F.
� pq ��
� mn �
G.m
:p�
n:q
H.m
:n�
p:q
3.W
rite
tw
o pr
opor
tion
s th
at m
atch
eac
h d
escr
ipti
on.
a.M
ean
s ar
e 5
and
8;ex
trem
es a
re 4
an
d 10
.
any
two
of
�4 5��
� 18 0�,�
4 8��
� 15 0�,�
1 50 ��
�8 4� ,�1 80 �
��5 4�
b.
Mea
ns
are
5 an
d 4;
extr
emes
are
pos
itiv
e in
tege
rs t
hat
are
dif
fere
nt
from
mea
ns.
any
two
of
�2 5��
� 14 0�,�
2 4��
� 15 0�,�
1 4��
� 25 0�,�
1 5��
� 24 0�,�
1 50 ��
�4 2� ,�1 40 �
��5 2� ,
�2 40 ��
�5 1� ,
or
�2 50 ��
�4 1�
Hel
pin
g Y
ou
Rem
emb
er
4.S
omet
imes
it
is e
asie
r to
rem
embe
r a
mat
hem
atic
al i
dea
if y
ou p
ut
it i
nto
wor
ds w
ith
out
usi
ng
any
mat
hem
atic
al s
ymbo
ls.H
ow c
an y
ou u
se t
his
app
roac
h t
o re
mem
ber
the
con
cept
of
equ
alit
y of
cro
ss p
rodu
cts?
Sam
ple
an
swer
:Th
e p
rod
uct
of
the
mea
ns
equ
als
the
pro
du
ct o
f th
e ex
trem
es.
©G
lenc
oe/M
cGra
w-H
ill30
0G
lenc
oe G
eom
etry
Go
lden
Rec
tan
gle
s
Use
a s
trai
ghte
dge
,com
pas
s,an
d t
he
inst
ruct
ion
s b
elow
to c
onst
ruct
a g
old
en r
ecta
ngl
e.
1.C
onst
ruct
squ
are
AB
CD
wit
h s
ides
of
2 cm
.
2.C
onst
ruct
th
e m
idpo
int
of A �
B�.C
all
the
mid
poin
t M
.
3.D
raw
AB
� �� .
Set
you
r co
mpa
ss a
t an
ope
nin
g eq
ual
to
MC
.U
se M
as t
he
cen
ter
to d
raw
an
arc
th
at i
nte
rsec
ts A
B� �
� .C
all
the
poin
t of
in
ters
ecti
on P
.
4.C
onst
ruct
a l
ine
thro
ugh
Pth
at i
s pe
rpen
dicu
lar
to A
B� �
� .
5.D
raw
DC
� ��
so t
hat
it
inte
rsec
ts t
he
perp
endi
cula
r li
ne
in s
tep
4.C
all
the
inte
rsec
tion
poi
nt
Q.A
PQ
Dis
a g
old
en r
ecta
ngl
ebe
cau
se t
he
rati
o of
its
len
gth
to
its
wid
th i
s 1.
618.
Ch
eck
this
con
clu
sion
by
fin
din
g th
e va
lue
of �Q A
PP �.R
ecta
ngl
es w
hos
e si
des
hav
e th
is r
atio
are
,it
is s
aid,
the
mos
t pl
easi
ng
to t
he
hu
man
eye
.
A f
igu
re c
onsi
stin
g of
sim
ilar
gol
den
rec
tan
gles
is
show
n b
elow
.Use
a c
omp
ass
and
th
e in
stru
ctio
ns
bel
ow t
o d
raw
qu
arte
r-ci
rcle
arc
s th
at f
orm
a s
pir
al l
ike
that
fou
nd
in
th
e sh
ell
of a
ch
amb
ered
nau
tilu
s.
6.U
sin
g A
as a
cen
ter,
draw
an
arc
th
at p
asse
s th
rou
gh
Ban
d C
.
7.U
sin
g D
as a
cen
ter,
draw
an
arc
th
at p
asse
s th
rou
gh
Can
d E
.
8.U
sin
g F
as a
cen
ter,
draw
an
arc
th
at p
asse
s th
rou
gh
Ean
d G
.
9.U
sin
g H
as a
cen
ter,
draw
an
arc
th
at p
asse
s th
rou
gh
Gan
d J
.
10.U
sin
g K
as a
cen
ter,
draw
an
arc
th
at p
asse
s th
rou
gh
Jan
d L
.
11.U
sin
g M
as a
cen
ter,
draw
an
arc
th
at p
asse
s th
rou
gh
Lan
d N
.
MN H
LF
D JK
BAC
G
E
AM
BP
DC
Q
Ch
eck
stu
den
ts’
con
stru
ctio
nm
arks
.Th
e fi
nal
fig
ure
sh
ou
ld lo
ok
like
the
on
e b
elo
w.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-1
6-1
Answers (Lesson 6-1)
© Glencoe/McGraw-Hill A5 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Sim
ilar
Po
lyg
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-2
6-2
©G
lenc
oe/M
cGra
w-H
ill30
1G
lenc
oe G
eom
etry
Lesson 6-2
Iden
tify
Sim
ilar
Fig
ure
s
Det
erm
ine
wh
eth
er t
he
tria
ngl
es
are
sim
ilar
.T
wo
poly
gon
s ar
e si
mil
ar i
f an
d on
ly i
f th
eir
corr
espo
ndi
ng
angl
es a
re c
ongr
uen
t an
d th
eir
corr
espo
ndi
ng
side
s ar
e pr
opor
tion
al.
�C
��
Zbe
cau
se t
hey
are
rig
ht
angl
es,a
nd
�B
��
X.
By
the
Th
ird
An
gle
Th
eore
m,�
A�
�Y
.
For
th
e si
des,
�B XC Z�
��2 20 3�
,�B X
A Y��
��2 20 3�
,an
d �A Y
C Z��
�2 20 3�.
Th
e si
de l
engt
hs
are
prop
orti
onal
.So
�B
CA
��
XZ
Y.
Is p
olyg
on W
XY
Z�
pol
ygon
PQ
RS
?
For
th
e si
des,
�W PQX �
��1 82 �
��3 2� ,
� QXY R�
��1 18 2�
��3 2� ,
� RYZ S�
��1 15 0�
��3 2� ,
and
�Z SW P��
�9 6��
�3 2� .S
o co
rres
pon
din
g si
des
are
prop
orti
onal
.
Als
o,�
W�
�P
,�X
��
Q,�
Y�
�R
,an
d �
Z�
�S
,so
corr
espo
ndi
ng
angl
es a
re c
ongr
uen
t.W
e ca
n c
oncl
ude
th
at
poly
gon
WX
YZ
�po
lygo
n P
QR
S.
Det
erm
ine
wh
eth
er e
ach
pai
r of
fig
ure
s is
sim
ilar
.If
they
are
sim
ilar
,giv
e th
era
tio
of c
orre
spon
din
g si
des
.
1.2.
yes;
�5 8�n
o
3.4.
yes;
�1 2�ye
s;�5 4�
10y
8y
37�
37�
116�
27�
5x
4x15
z12
z
2211
1610 equi
late
ral t
riang
les
15
9
12
18XW
ZY
10
6
8
12
QP S
R
20�
2�� 23
�2�
2023
2320
20�
�223
��2
45�
45�
CA
XZ
BY
Exer
cises
Exer
cises
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
©G
lenc
oe/M
cGra
w-H
ill30
2G
lenc
oe G
eom
etry
Scal
e Fa
cto
rsW
hen
tw
o po
lygo
ns
are
sim
ilar
,th
e ra
tio
of t
he
len
gth
s of
cor
resp
ondi
ng
side
s is
cal
led
the
scal
e fa
ctor
.At
the
righ
t,�
AB
C�
�X
YZ
.Th
e sc
ale
fact
or o
f �
AB
Cto
�X
YZ
is 2
an
d th
e sc
ale
fact
or o
f
�X
YZ
to �
AB
Cis
�1 2� .3
cm
6 cm
8 cm
10 c
m5
cm4
cm ZY
XCA
B
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Sim
ilar
Po
lyg
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-2
6-2
Th
e tw
o p
olyg
ons
are
sim
ilar
.Fin
d x
and
y.
Use
th
e co
ngr
uen
t an
gles
to
wri
te t
he
corr
espo
ndi
ng
vert
ices
in
ord
er.
�R
ST
��
MN
PW
rite
pro
port
ion
s to
fin
d x
and
y.
�3 12 6��
� 1x 3��3 y8 �
��3 12 6�
16x
�32
(13)
32y
�38
(16)
x�
26y
�19
x
y38
3216
13T
S
RP
N
M
�A
BC
��
CD
E.F
ind
th
e sc
ale
fact
or a
nd
fin
d t
he
len
gth
s of
C �D�
and
D�E�
.
AC
�3
�0
�3
and
CE
�9
�3
�6.
Th
esc
ale
fact
or o
f �
CD
Eto
�A
BC
is 6
:3 o
r 2:
1.U
sin
g th
e D
ista
nce
For
mu
la,
AB
��
1 �
9�
��
10�an
d B
C�
�4
�9
��
�13�
.Th
e le
ngt
hs
of t
he
side
s of
�C
DE
are
twic
e th
ose
of �
AB
C,
so D
C�
2(B
A)
or 2
�10�
and
DE
�2(
BC
) or
2�
13�.
x
y
A( 0
, 0)
B( 1
, 3) C( 3
, 0)
E( 9
, 0)
D
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Eac
h p
air
of p
olyg
ons
is s
imil
ar.F
ind
xan
d y
.
1.2.
x�
6;y
�15
x�
2.25
;y
�5
3.4.
x�
27;
y�
15x
�24
;y
�27
5.In
Exa
mpl
e 2
abov
e,po
int
Dh
as c
oord
inat
es (
5,6)
.Use
th
e D
ista
nce
For
mu
la t
o ve
rify
the
len
gth
s of
C �D�
and
D�E�
.
CD
�2�
10�,D
E�
2�13�
40
36x
30
18y
12
y �
1
18
24
x18
10y
4.5
x2.
5 5
9
12y8 10
x
Answers (Lesson 6-2)
© Glencoe/McGraw-Hill A6 Glencoe Geometry
Skil
ls P
ract
ice
Sim
ilar
Po
lyg
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-2
6-2
©G
lenc
oe/M
cGra
w-H
ill30
3G
lenc
oe G
eom
etry
Lesson 6-2
Det
erm
ine
wh
eth
er e
ach
pai
r of
fig
ure
s is
sim
ilar
.Ju
stif
y yo
ur
answ
er.
1.2.
�A
BC
��
DE
F;
�A
��
D,
rho
mbu
s P
QR
S�
rho
mbu
s W
XY
Z;�
C�
�F
,an
d �
B�
�E
�P
��
W,�
Q�
�X
,�R
��
Y,
bec
ause
if t
wo
an
gle
s o
f o
ne
�S
��
Z;
tria
ng
le a
re c
on
gru
ent
to t
wo
� WP
Q X��
�Q XYR �
��R Y
S Z��
� ZS WP ��
� 73 .5��
0.4
ang
les
of
a se
con
d t
rian
gle
,th
en
the
thir
d a
ng
les
are
con
gru
ent;
�A DB E�
��B E
C F��
�C FDA �
��3 2� .
Eac
h p
air
of p
olyg
ons
is s
imil
ar.W
rite
a s
imil
arit
y st
atem
ent,
and
fin
d x
,th
em
easu
re(s
) of
in
dic
ated
sid
e(s)
,an
d t
he
scal
e fa
ctor
.
3.G �
H�4.
S�T�
and
S�U�
AB
CD
�E
FG
H;
�W
XY
��
ST
U(o
r �
SU
T)
6�1 2� ;
6�1 2� ;
27;
12;
12;
�1 3�
5.W�
T�6.
T�S�
and
S�P�
PQ
RS
�U
VW
T;
�L
MN
��
TS
P;
5;6;
�3 2�13
;12
;15
;�2 3�
10
8
x �
2
x �
1
SP
MN
L
T
9
10 3
x �
1
5
P
S
RU
V
WT
Q
x �
5
x �
5
9
3
44
S
W
XY
T U2614
1213
BC
DA
1376
x G
EH
F
7.5
7.5
7.5
7.5
ZY
XW
3
3
33
SR
QP
10.5
69
59�
35�
AC
B
7
46
59�
35�
DF
E
©G
lenc
oe/M
cGra
w-H
ill30
4G
lenc
oe G
eom
etry
Det
erm
ine
wh
eth
er e
ach
pai
r of
fig
ure
s is
sim
ilar
.Ju
stif
y yo
ur
answ
er.
1.2.
JKL
M�
QR
SP
;�
AB
C�
�U
VT
;�
A�
�U
,�
J�
�Q
,�K
��
R,
�B
��
V,a
nd
�C
��
Tby
th
e �
L�
�S
,�M
��
P,a
nd
T
hir
d A
ng
le T
heo
rem
an
d
� QJKR�
�� RK
SL ��
�L SM P��
� PMQJ �
��5 3�
or
1.67
�A UB V�
��B V
C T��
�C TUA �
��2 3� .
Eac
h p
air
of p
olyg
ons
is s
imil
ar.W
rite
a s
imil
arit
y st
atem
ent,
and
fin
d x
,th
em
easu
re(s
) of
in
dic
ated
sid
e(s)
,an
d t
he
scal
e fa
ctor
.
3.L �
M�an
d M�
N�4.
D�E�
and
D�F�
AB
CD
�L
MN
P;
�A
BC
��
DE
F;
�3 2� ;10
�1 2� ;7�
1 2� ;�4 3�
7;4;
8;�3 2�
5.C
OO
RD
INA
TE G
EOM
ETRY
Tri
angl
e A
BC
has
ver
tice
s A
(0,0
),B
(�4,
0),a
nd
C(�
2,4)
.T
he
coor
din
ates
of
each
ver
tex
are
mu
ltip
lied
by
3 to
cre
ate
�A
EF
.Sh
ow t
hat
�A
EF
issi
mil
ar t
o �
AB
C.
AB
�4,
BC
�C
A�
2�5�
and
AE
�12
,EF
�FA
�6�
5�.�A A
B E��
�B EC F�
��C FA
A ��
�1 3� ;�
A�
�A
and
sin
ce B�
C�|| E�
F�,�
B�
�E
and
�C
��
F.S
ince
th
e co
rres
po
nd
ing
an
gle
s ar
e co
ng
ruen
t an
d t
he
corr
esp
on
din
g s
ides
are
pro
po
rtio
nal
,�A
BC
��
AE
F.
6.IN
TER
IOR
DES
IGN
Gra
ham
use
d th
e sc
ale
draw
ing
of h
is l
ivin
g ro
om t
o de
cide
wh
ere
to p
lace
fu
rnit
ure
.Fin
d th
e di
men
sion
s of
th
eli
vin
g ro
om i
f th
e sc
ale
in t
he
draw
ing
is 1
in
ch �
4.5
feet
.18
ft
by 1
1 ft
3 in
.4
in.
21 – 2 in.
612
40�
40�
x �
1
x �
3A
F
E
D
BC
14
10
x �
6
x �
9
CN
MD
AB
LP
2412
1416
2118
VU
AC
BT
2512
9
14.4
1524
20
15
JM
QRS
PK
LPra
ctic
e (
Ave
rag
e)
Sim
ilar
Po
lyg
on
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-2
6-2
Answers (Lesson 6-2)
© Glencoe/McGraw-Hill A7 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csS
imila
r P
oly
go
ns
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-2
6-2
©G
lenc
oe/M
cGra
w-H
ill30
5G
lenc
oe G
eom
etry
Lesson 6-2
Pre-
Act
ivit
yH
ow d
o ar
tist
s u
se g
eom
etri
c p
atte
rns?
Rea
d th
e in
trod
uct
ion
to
Les
son
6-2
at
the
top
of p
age
289
in y
our
text
book
.•
Des
crib
e th
e fi
gure
s th
at h
ave
sim
ilar
sh
apes
. Sam
ple
an
swer
:L
igh
t an
d d
ark
fig
ure
s ar
e ar
ran
ged
in a
lter
nat
ing
pat
tern
s.F
igu
res
that
hav
e th
eir
feet
po
inti
ng
to
war
d t
he
cen
ter
of
the
circ
le h
ave
the
sam
e sh
ape
but
dif
fere
nt
size
s.•
Wh
at h
appe
ns
to t
he
figu
res
as y
our
eyes
mov
e fr
om t
he
cen
ter
to t
he
oute
r ed
ge?
Sam
ple
an
swer
:Th
e fi
gu
res
bec
om
e sm
alle
r.
Rea
din
g t
he
Less
on
1.C
ompl
ete
each
sen
ten
ce.
a.T
wo
poly
gon
s th
at h
ave
exac
tly
the
sam
e sh
ape,
but
not
nec
essa
rily
th
e sa
me
size
,ar
e .
b.
Tw
o po
lygo
ns
are
con
gru
ent
if t
hey
hav
e ex
actl
y th
e sa
me
shap
e an
d th
e sa
me
.c.
Tw
o po
lygo
ns
are
sim
ilar
if
thei
r co
rres
pon
din
g an
gles
are
an
d th
eir
corr
espo
ndi
ng
side
s ar
e .
d.
Tw
o po
lygo
ns
are
con
gru
ent
if t
hei
r co
rres
pon
din
g an
gles
are
an
d th
eir
corr
espo
ndi
ng
side
s ar
e .
e.T
he
rati
o of
th
e le
ngt
hs
of c
orre
spon
din
g si
des
of t
wo
sim
ilar
fig
ure
s is
cal
led
the
.f.
Mu
ltip
lyin
g th
e co
ordi
nat
es o
f al
l po
ints
of
a fi
gure
in
th
e co
ordi
nat
e pl
ane
by a
sca
le
fact
or t
o ge
t a
sim
ilar
fig
ure
is
call
ed a
.
g.If
tw
o po
lygo
ns a
re s
imila
r w
ith
a sc
ale
fact
or o
f 1,
then
the
pol
ygon
s ar
e .
2.D
eter
min
e w
het
her
eac
h s
tate
men
t is
alw
ays,
som
etim
es,o
r n
ever
tru
e.a.
Tw
o si
mil
ar t
rian
gles
are
con
gru
ent.
som
etim
esb
.T
wo
equ
ilat
eral
tri
angl
es a
re c
ongr
uen
t.so
met
imes
c.A
n e
quil
ater
al t
rian
gle
is s
imil
ar t
o a
scal
ene
tria
ngl
e.n
ever
d.
Tw
o re
ctan
gles
are
sim
ilar
.so
met
imes
e.T
wo
isos
cele
s ri
ght
tria
ngl
es a
re c
ongr
uen
t.so
met
imes
f.T
wo
isos
cele
s ri
ght
tria
ngl
es a
re s
imil
ar.
alw
ays
g.A
squ
are
is s
imil
ar t
o an
equ
ilat
eral
tri
angl
e.n
ever
h.
Tw
o ac
ute
tri
angl
es a
re s
imil
ar.
som
etim
esi.
Tw
o re
ctan
gles
in
wh
ich
th
e le
ngt
h i
s tw
ice
the
wid
th a
re s
imil
ar.
alw
ays
j.T
wo
con
gru
ent
poly
gon
s ar
e si
mil
ar.
alw
ays
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al v
ocab
ula
ry t
erm
is
to r
elat
e it
to
wor
dsu
sed
in e
very
day
life
.Th
e w
ord
scal
eh
as m
any
mea
nin
gs i
n E
ngl
ish
.Giv
e th
ree
phra
ses
that
in
clu
de t
he
wor
d sc
ale
in a
way
th
at i
s re
late
d to
pro
port
ion
s.S
amp
le a
nsw
er:
scal
e d
raw
ing
,sca
le m
od
el,s
cale
of
mile
s (o
n a
map
)
con
gru
ent
dila
tio
n
scal
e fa
cto
r
con
gru
ent
con
gru
ent
pro
po
rtio
nal
con
gru
ent
sizesi
mila
r
©G
lenc
oe/M
cGra
w-H
ill30
6G
lenc
oe G
eom
etry
Co
nst
ruct
ing
Sim
ilar
Po
lyg
on
sH
ere
are
fou
r st
eps
for
con
stru
ctin
g a
poly
gon
th
at i
s si
mil
ar t
o an
d w
ith
side
s tw
ice
as l
ong
as t
hos
e of
an
exi
stin
g po
lygo
n.
Ste
p 1
Ch
oose
an
y po
int
eith
er i
nsi
de o
r ou
tsid
e th
e po
lygo
n a
nd
labe
l it
O.
Ste
p 2
Dra
w r
ays
from
Oth
rou
gh e
ach
ver
tex
of t
he
poly
gon
.S
tep
3F
or v
erte
x V
,set
th
e co
mpa
ss t
o le
ngt
h O
V.T
hen
loc
ate
a n
ew p
oin
tV
�on
ray
OV
such
th
at V
V�
�O
V.T
hu
s,O
V�
�2(
OV
).S
tep
4R
epea
t S
tep
3 fo
r ea
ch v
erte
x.C
onn
ect
poin
ts V
�,W
�,X
�an
d Y
�to
form
th
e n
ew p
olyg
on.
Tw
o co
nst
ruct
ion
s of
pol
ygon
s si
mil
ar t
o an
d w
ith
sid
es t
wic
e th
ose
of V
WX
Yar
e sh
own
bel
ow.N
otic
e th
at t
he
plac
emen
t of
poi
nt
Odo
es n
ot a
ffec
t th
e si
zeor
sh
ape
of V
�W�X
�Y�,
only
its
loc
atio
n.
Tra
ce e
ach
pol
ygon
.Th
en c
onst
ruct
a s
imil
ar p
olyg
on w
ith
sid
es t
wic
e as
lon
g as
thos
e of
th
e gi
ven
pol
ygon
.S
ee s
tud
ents
’co
nst
ruct
ion
s.
1.2.
DE
F
GC
A
B
VW
YX
VW
YX
V�
W�
X�
Y�
O
VW
YX
V�
W�
X�
Y�
O
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-2
6-2
3.E
xpla
in h
ow t
o co
nst
ruct
a s
imil
arpo
lygo
n w
ith
side
s th
ree
tim
es t
he l
engt
hof
th
ose
of p
olyg
on H
IJK
L.T
hen
do
the
con
stru
ctio
n.
See
stu
den
ts’w
ork
.
4.E
xpla
in h
ow t
o co
nst
ruct
a s
imil
ar
poly
gon
1�1 2�
tim
es t
he
len
gth
of
thos
e of
poly
gon
MN
PQ
RS
.Th
en d
o th
eco
nst
ruct
ion
.S
ee s
tud
ents
’wo
rk.
M
S
R
N
Q
P
HI
LK
J
Answers (Lesson 6-2)
© Glencoe/McGraw-Hill A8 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Sim
ilar T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-3
6-3
©G
lenc
oe/M
cGra
w-H
ill30
7G
lenc
oe G
eom
etry
Lesson 6-3
Iden
tify
Sim
ilar
Tria
ng
les
Her
e ar
e th
ree
way
s to
sh
ow t
hat
tw
o tr
ian
gles
are
sim
ilar
.
AA
Sim
ilari
tyTw
o an
gles
of
one
tria
ngle
are
con
grue
nt t
o tw
o an
gles
of
anot
her
tria
ngle
.
SS
S S
imila
rity
The
mea
sure
s of
the
cor
resp
ondi
ng s
ides
of
two
tria
ngle
s ar
e pr
opor
tiona
l.
SA
S S
imila
rity
The
mea
sure
s of
tw
o si
des
of o
ne t
riang
le a
re p
ropo
rtio
nal t
o th
e m
easu
res
of
two
corr
espo
ndin
g si
des
of a
noth
er t
riang
le,
and
the
incl
uded
ang
les
are
cong
ruen
t.
Det
erm
ine
wh
eth
er t
he
tria
ngl
es a
re s
imil
ar.
� DAC F�
��6 9�
��2 3�
�B EC F�
�� 18 2�
��2 3�
� DAB E�
��1 10 5�
��2 3�
�A
BC
��
DE
Fby
SS
S S
imil
arit
y.
15
129
DE
F
10
86
AB
C
Det
erm
ine
wh
eth
er t
he
tria
ngl
es a
re s
imil
ar.
�3 4��
�6 8� ,so
�M QRN �
��N R
SP �.
m�
N�
m�
R,s
o �
N�
�R
.�
NM
P�
�R
QS
by S
AS
Sim
ilar
ity.
8
4 70�Q
RS
6
3 70�
M
NP
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Det
erm
ine
wh
eth
er e
ach
pai
r of
tri
angl
es i
s si
mil
ar.J
ust
ify
you
r an
swer
.
1.2.
yes;
AA
Sim
ilari
tyn
o;
�3 26 4��
�2 10 8�
3.4.
yes;
AA
Sim
ilari
tyye
s;S
AS
Sim
ilari
ty
5.6.
yes;
SA
S S
imila
rity
yes;
SS
S S
imila
rity
32
20
1540
2524
3926
1624
18
8 65�
9
4 65�
36
2018
24
©G
lenc
oe/M
cGra
w-H
ill30
8G
lenc
oe G
eom
etry
Use
Sim
ilar
Tria
ng
les
Sim
ilar
tri
angl
es c
an b
e u
sed
to f
ind
mea
sure
men
ts.
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Sim
ilar T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-3
6-3
�A
BC
��
DE
F.
Fin
d x
and
y.
� DAC F�
��B E
C F�� DA
B E��
�B EC F�
��1 98 �
� 1y 8��
�1 98 �
18x
�9(
18�
3�)9y
�32
4x
�9 �
3�y
�36
18�
3��
x
18�
�3
B CA
1818
9y
x
E FD
A p
erso
n 6
fee
t ta
ll c
asts
a 1.
5-fo
ot-l
ong
shad
ow a
t th
e sa
me
tim
eth
at a
fla
gpol
e ca
sts
a 7-
foot
-lon
gsh
adow
.How
tal
l is
th
e fl
agp
ole?
Th
e su
n’s
ray
s fo
rm s
imil
ar t
rian
gles
.U
sin
g x
for
the
hei
ght
of t
he
pole
,�6 x�
��1 7.5 �
,so
1.5
x�
42 a
nd
x�
28.
Th
e fl
agpo
le i
s 28
fee
t ta
ll.
7 ft
6 ft
?
1.5
ft
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Eac
h p
air
of t
rian
gles
is
sim
ilar
.Fin
d x
and
y.
1.2.
x�
26;
y�
17.5
x�
24;
y�
30
3.4.
x�
12;
y�
24�
2�x
�19
.3;
y�
46
5.6.
x�
18;
y�
10.8
x�
10�2 3� ;
y�
16�1 2�
7.T
he
hei
ghts
of
two
vert
ical
pos
ts a
re 2
met
ers
and
0.45
met
er.W
hen
th
e sh
orte
r po
stca
sts
a sh
adow
th
at i
s 0.
85 m
eter
lon
g,w
hat
is
the
len
gth
of
the
lon
ger
post
’s s
had
ow
to t
he
nea
rest
hu
ndr
edth
?3.
78 m
y
x32
10
2230
yx
9
87.2
16
yx
6038
.623
3024
36�
�2y
x
3636
20 13
26y
x �
235
13
1020
y
x
Answers (Lesson 6-3)
© Glencoe/McGraw-Hill A9 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Sim
ilar T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-3
6-3
©G
lenc
oe/M
cGra
w-H
ill30
9G
lenc
oe G
eom
etry
Lesson 6-3
Det
erm
ine
wh
eth
er e
ach
pai
r of
tri
angl
es i
s si
mil
ar.J
ust
ify
you
r an
swer
.
1.2.
No
;th
e si
des
are
no
t p
rop
ort
ion
al.
yes;
�A
BC
��
PQ
R(o
r �
QP
R);
SS
S S
imila
rity
3.4.
yes;
�S
TU
��
JPM
;ye
s;�
SK
M�
�R
TQ
;S
AS
Sim
ilari
tyA
A S
imila
rity
ALG
EBR
AId
enti
fy t
he
sim
ilar
tri
angl
es,a
nd
fin
d x
and
th
e m
easu
res
of t
he
ind
icat
ed s
ides
.
5.A �
C�an
d E�
D�6.
J�L�an
d L�
M�
�A
BC
��
DB
E;
15;
16;
20�
JKL
��
MN
L;
10;
28;
7
7.E�
H�an
d E�
F�8.
U�T�
and
R�T�
�D
EF
��
GE
H;
4;9;
18�
RS
T�
�U
VT
;12
;6;
14
6
14
S
U
V
RT
x �
4 x �
612
9
6
9
D
E
FH G
x �
5
16
4MN
L
J
K
x �
18
x �
3
1512
D
EC
BAx
� 1
x �
5
30�
60�
RQM
S
K
T
2170
�
15
MJ
P
1470
�10
U
S
T
12
128
9
96
CP
QR
A
B
2014
1312
219 S
WX
Y
R
T
©G
lenc
oe/M
cGra
w-H
ill31
0G
lenc
oe G
eom
etry
Det
erm
ine
wh
eth
er e
ach
pai
r of
tri
angl
es i
s si
mil
ar.J
ust
ify
you
r an
swer
.
1.2.
yes;
�JA
K�
�W
SY
;N
o;
the
sid
es a
re n
ot
pro
po
rtio
nal
.S
AS
Sim
ilari
ty
ALG
EBR
AId
enti
fy t
he
sim
ilar
tri
angl
es,a
nd
fin
d x
and
th
e m
easu
res
of t
he
ind
icat
ed s
ides
.
3.L �
M�an
d Q�
P�4.
N�L�
and
M�L�
�L
MN
��
PQ
N;
9;12
;8
�JL
N�
�K
LM
;2;
21;
14
Use
th
e gi
ven
in
form
atio
n t
o fi
nd
eac
h m
easu
re.
5.If
T�S�
|| Q�R�
,TS
�6,
PS
�x
�7,
6.If
E�F�
|| H�I�,
EF
�3,
EG
�x
�1,
QR
�8,
and
SR
�x
�1,
HI
�4,
and
HG
�x
�3,
fin
d P
San
d P
R.
fin
d E
Gan
d H
G.
PS
�12
;P
R�
16E
G�
6;H
G�
8
IND
IREC
T M
EASU
REM
ENT
For
Exe
rcis
es 7
an
d 8
,use
th
e fo
llow
ing
info
rmat
ion
.A
lig
hth
ouse
cas
ts a
128
-foo
t sh
adow
.A n
earb
y la
mpp
ost
that
mea
sure
s 5
feet
3 i
nch
es c
asts
an 8
-foo
t sh
adow
.
7.W
rite
a p
ropo
rtio
n t
hat
can
be
use
d to
det
erm
ine
the
hei
ght
of t
he
ligh
thou
se.
Sam
ple
an
swer
:If
x�
hei
gh
t o
f lig
hth
ou
se,�
5.x 25�
��12 88 �
.
8.W
hat
is
the
hei
ght
of t
he
ligh
thou
se?
84 f
t
F
GE
H
I
SR
P
QT
8N
JL
K
Mx
� 5
6x �
2
24
12
18N
P
QL
M
x �
3x
� 1
18
1612
.516
14
11
M
LN
SR
T42
�
42�
24
1812
16J
AK
YS
W
Pra
ctic
e (
Ave
rag
e)
Sim
ilar T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-3
6-3
Answers (Lesson 6-3)
© Glencoe/McGraw-Hill A10 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csS
imila
r Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-3
6-3
©G
lenc
oe/M
cGra
w-H
ill31
1G
lenc
oe G
eom
etry
Lesson 6-3
Pre-
Act
ivit
yH
ow d
o en
gin
eers
use
geo
met
ry?
Rea
d th
e in
trod
uct
ion
to
Les
son
6-3
at
the
top
of p
age
298
in y
our
text
book
.•
Wh
at d
oes
it m
ean
to
say
that
tri
angu
lar
shap
es r
esu
lt i
n r
igid
con
stru
ctio
n?
Sam
ple
an
swer
:A
tri
ang
ula
r st
ruct
ure
will
no
tch
ang
e it
s sh
ape
or
size
if y
ou
pu
sh o
n it
s si
des
wit
ho
ut
bre
akin
g o
r b
end
ing
th
em.
•W
hat
wou
ld h
appe
n i
f th
e sh
apes
use
d in
th
e co
nst
ruct
ion
wer
equ
adri
late
rals
?S
amp
le a
nsw
er:T
he
stru
ctu
re m
igh
t co
llap
se,
bec
ause
qu
adri
late
rals
are
no
t ri
gid
.
Rea
din
g t
he
Less
on
1.S
tate
wh
eth
er e
ach
con
diti
on g
uar
ante
es t
hat
tw
o tr
ian
gles
are
con
gru
ent
or s
imil
ar.
If t
he
con
diti
on g
uar
ante
es t
hat
th
e tr
ian
gles
are
bot
h s
imil
ar a
nd
con
gru
ent,
wri
teco
ngr
uen
t.If
th
ere
is n
ot e
nou
gh i
nfo
rmat
ion
to
guar
ante
e th
at t
he
tria
ngl
es w
ill
beco
ngr
uen
t or
sim
ilar
,wri
te n
eith
er.
a.T
wo
side
s an
d th
e in
clu
ded
angl
e of
on
e tr
ian
gle
are
con
gru
ent
to t
wo
side
s an
d th
ein
clu
ded
angl
e of
th
e ot
her
tri
angl
e.co
ng
ruen
tb
.T
he
mea
sure
s of
all
th
ree
pair
s of
cor
resp
ondi
ng
side
s ar
e pr
opor
tion
al.
sim
ilar
c.T
wo
angl
es o
f on
e tr
ian
gle
are
con
gru
ent
to t
wo
angl
es o
f th
e ot
her
tri
angl
e.si
mila
rd
.T
wo
angl
es a
nd
a n
onin
clu
ded
side
of
one
tria
ngl
e ar
e co
ngr
uen
t to
tw
o an
gles
an
dth
e co
rres
pon
din
g n
onin
clu
ded
side
of
the
oth
er t
rian
gle.
con
gru
ent
e.T
he
mea
sure
s of
tw
o si
des
of a
tri
angl
e ar
e pr
opor
tion
al t
o th
e m
easu
res
of t
wo
corr
espo
ndi
ng
side
s of
an
oth
er t
rian
gle,
and
the
incl
ude
d an
gles
are
con
gru
ent.
sim
ilar
f.T
he
thre
e si
des
of o
ne
tria
ngl
e ar
e co
ngr
uen
t to
th
e th
ree
side
s of
th
e ot
her
tri
angl
e.co
ng
ruen
tg.
The
thr
ee a
ngle
s of
one
tri
angl
e ar
e co
ngru
ent
to t
he t
hree
ang
les
of t
he o
ther
tri
angl
e.si
mila
rh
.O
ne
acu
te a
ngl
e of
a r
igh
t tr
ian
gle
is c
ongr
uen
t to
on
e ac
ute
an
gle
of a
not
her
rig
ht
tria
ngl
e.si
mila
ri.
Th
e m
easu
res
of t
wo
side
s of
a t
rian
gle
are
prop
orti
onal
to
the
mea
sure
s of
tw
o si
des
of a
not
her
tri
angl
e.n
eith
er2.
Iden
tify
eac
h of
the
fol
low
ing
as a
n ex
ampl
e of
a r
efle
xive
,sym
met
ric,
or t
rans
itiv
epr
oper
ty.
a.If
�R
ST
��
UV
W,t
hen
�U
VW
��
RS
T.
sym
met
ric
b.
If �
RS
T�
�U
VW
and
�U
VW
��
OP
Q,t
hen
�R
ST
��
OP
Q.
tran
siti
vec.
�R
ST
��
RS
Tre
flex
ive
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r so
met
hin
g is
to
expl
ain
it
to s
omeo
ne
else
.Su
ppos
e on
e of
you
rcl
assm
ates
is
hav
ing
trou
ble
un
ders
tan
din
g th
e di
ffer
ence
bet
wee
n S
AS
for
con
gru
ent
tria
ngl
es a
nd
SA
S f
or s
imil
ar t
rian
gles
.How
can
you
exp
lain
th
e di
ffer
ence
to
him
?S
amp
le a
nsw
er:
In S
AS
fo
r co
ng
ruen
ce,c
orr
esp
on
din
g s
ides
are
con
gru
ent,
wh
ile in
SA
S f
or
sim
ilari
ty,c
orr
esp
on
din
g s
ides
are
pro
po
rtio
nal
.(In
bo
th,t
he
incl
ud
ed a
ng
les
are
con
gru
ent.
)
©G
lenc
oe/M
cGra
w-H
ill31
2G
lenc
oe G
eom
etry
Rat
io P
uzz
les
wit
h T
rian
gle
sIf
you
kn
ow t
he
peri
met
er o
f a
tria
ngl
e an
d th
e ra
tios
of
the
side
s,yo
u c
anfi
nd
the
len
gth
s of
th
e si
des.
Th
e p
erim
eter
of
a tr
ian
gle
is 8
4 u
nit
s.T
he
sid
es h
ave
len
gth
s r,
s,an
d t
.Th
e ra
tio
of s
to r
is 5
:3,a
nd
th
e ra
tio
of t
to r
is2
:1.F
ind
th
e le
ngt
h o
f ea
ch s
ide.
Sin
ce b
oth
rat
ios
con
tain
r,r
ewri
te o
ne
or b
oth
rat
ios
to m
ake
rth
e sa
me.
You
can
wri
te t
he
rati
o of
tto
ras
6:3
.Now
you
can
wri
te a
th
ree-
part
rat
io.
r:s:
t�
3:5
:6
Th
ere
is a
nu
mbe
r x
such
th
at r
�3
x,s
�5
x,an
d t
�6
x.S
ince
you
kn
ow t
he
peri
met
er,8
4,yo
u c
an u
se a
lgeb
ra t
o fi
nd
the
len
gth
s of
th
e si
des.
r�
s�
t�
843
x�
5x�
6x�
8414
x�
84x
�6
3x
�18
,5x
�30
,6x
�36
So
r�
18,s
�30
,an
d t
�36
.
Fin
d t
he
len
gth
s of
th
e si
des
of
each
tri
angl
e.
1.T
he
peri
met
er o
f a
tria
ngl
e is
75
un
its.
Th
e si
des
hav
e le
ngt
hs
a,b,
and
c.T
he
rati
o of
bto
ais
3:5
,an
d th
e ra
tio
of c
to a
is 7
:5.F
ind
the
len
gth
of
each
sid
e.
a�
25,b
�15
,c�
35
2.T
he
peri
met
er o
f a
tria
ngl
e is
88
un
its.
Th
e si
des
hav
e le
ngt
hs
d,e
,an
d f.
Th
e ra
tio
of e
to d
is 3
:1,a
nd
the
rati
o of
fto
eis
10:
9.F
ind
the
len
gth
of
each
sid
e.
d�
12,e
�36
,f�
40
3.T
he
peri
met
er o
f a
tria
ngl
e is
91
un
its.
Th
e si
des
hav
e le
ngt
hs
p,q,
and
r.T
he
rati
o of
pto
ris
3:1
,an
d th
e ra
tio
of q
to r
is 5
:2.F
ind
the
len
gth
of
each
sid
e.
p�
42,q
�35
,r�
14
4.T
he
peri
met
er o
f a
tria
ngl
e is
68
un
its.
Th
e si
des
hav
e le
ngt
hs
g,h
,an
d j.
Th
e ra
tio
of j
to g
is 2
:1,a
nd
the
rati
o of
hto
gis
5:4
.Fin
d th
e le
ngt
h o
f ea
ch s
ide.
g�
16,h
�20
,j�
32
5.W
rite
a p
robl
em s
imil
ar t
o th
ose
abov
e in
volv
ing
rati
os i
n t
rian
gles
.
See
stu
den
ts’w
ork
.
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-3
6-3
Exam
ple
Exam
ple
Answers (Lesson 6-3)
© Glencoe/McGraw-Hill A11 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Par
alle
l Lin
es a
nd
Pro
po
rtio
nal
Par
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-4
6-4
©G
lenc
oe/M
cGra
w-H
ill31
3G
lenc
oe G
eom
etry
Lesson 6-4
Pro
po
rtio
nal
Par
ts o
f Tr
ian
gle
sIn
an
y tr
ian
gle,
a li
ne
para
llel
to
one
side
of
a tr
ian
gle
sepa
rate
s th
e ot
her
tw
o si
des
prop
orti
onal
ly.T
he
con
vers
e is
als
o tr
ue.
If X
and
Yar
e th
e m
idpo
ints
of
R �T�
and
S�T�
,th
en X�
Y�is
am
idse
gmen
tof
th
e tr
ian
gle.
Th
e T
rian
gle
Mid
segm
ent
Th
eore
mst
ates
th
at a
mid
segm
ent
is p
aral
lel
to t
he
thir
d si
de a
nd
is h
alf
its
len
gth
.
If X
Y� �
�|| R
S��
� ,th
en �R X
TX ��
� YSY T�
.
If �R X
TX ��
� YSY T�
,th
en X
Y� �
�|| R
S� �
� .
If X�
Y�is
a m
idse
gmen
t,th
en X
Y� �
�|| R
S� �
�an
d X
Y�
�1 2� RS
.
YS
T
R
X
In �
AB
C,E�
F�|| C�
B�.F
ind
x.
Sin
ce E�
F�|| C�
B�,�
FABF �
�� EA
E C�.
�x x� �2 22
��
�1 68 �
6x�
132
�18
x�
3696
�12
x8
�x
F
C
BA
18
x �
22
x �
2
6E
A t
rian
gle
has
ver
tice
sD
(3,6
),E
(�3,
�2)
,an
d F
(7,�
2).
Mid
segm
ent
G �H�
is p
aral
lel
to E�
F�.F
ind
th
e le
ngt
h o
f G �
H�.
G �H�
is a
mid
segm
ent,
so i
ts l
engt
h i
s on
e-h
alf
that
of
E �F�
.Poi
nts
Ean
d F
hav
e th
esa
me
y-co
ordi
nat
e,so
EF
�7
�(�
3) �
10.
Th
e le
ngt
h o
f m
idse
gmen
t G �
H�is
5.
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d x
.
1.7
2.10
3.17
.5
4.8
5.12
6.5
7.In
Exa
mpl
e 2,
fin
d th
e sl
ope
of E�
F�an
d sh
ow t
hat
E�F�
|| G�H�
.
Th
e en
dp
oin
ts o
f G�
H�ar
e at
(0,
2) a
nd
(5,
2).T
he
slo
pe
of
G�H�
is 0
an
d t
he
slo
pe
of
E�F�
is 0
.G�H�
and
E�F�
are
par
alle
l.
3010
x �
10
x11
x �
12
x33
3010
24x
35x9
20
x
185
5x
7
©G
lenc
oe/M
cGra
w-H
ill31
4G
lenc
oe G
eom
etry
Div
ide
Seg
men
ts P
rop
ort
ion
ally
Wh
en
thre
e or
mor
e pa
rall
el l
ines
cu
t tw
o tr
ansv
ersa
ls,
they
sep
arat
e th
e tr
ansv
ersa
ls i
nto
pro
port
ion
al
part
s.If
th
e ra
tio
of t
he
part
s is
1,t
hen
th
e pa
rall
elli
nes
sep
arat
e th
e tr
ansv
ersa
ls i
nto
con
gru
ent
part
s.
If �
1|| �
2|| �
3,If
�4
|| �5
|| �6
and
then
�a b��
� dc � .�u v�
�1,
then
�w x��
1.
Ref
er t
o li
nes
�1,
� 2,a
nd
�3
abov
e.If
a�
3,b
�8,
and
c�
5,fi
nd
d.
� 1|| �
2|| �
3so
�3 8��
� d5 � .T
hen
3d
�40
an
d d
�13
�1 3� .
Fin
d x
and
y.
1.2.
x�
8x
�9
3.4.
x�
5;y
�2
y�
3�1 3�
5.6.
x�
12y
�3
1632
y �
3y
34
x �
4x
5
8y
� 2
y2x
� 4
3 x �
12y
� 2
3 y
2x �
6
x �
3
12
x �
12
123x5x
a
bd
uv
xw
ct
n m
s� 1
� 4� 5
� 6
� 2
� 3
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Par
alle
l Lin
es a
nd
Pro
po
rtio
nal
Par
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-4
6-4
Exam
ple
Exam
ple
Exer
cises
Exer
cises
Answers (Lesson 6-4)
© Glencoe/McGraw-Hill A12 Glencoe Geometry
Skil
ls P
ract
ice
Par
alle
l Lin
es a
nd
Pro
po
rtio
nal
Par
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-4
6-4
©G
lenc
oe/M
cGra
w-H
ill31
5G
lenc
oe G
eom
etry
Lesson 6-4
1.If
JK
�7,
KH
�21
,an
d J
L�
6,2.
Fin
d x
and
TV
if R
U�
8,U
S�
14,
fin
d L
I.T
V�
x�
1 an
d V
S�
17.5
.
1811
;10
Det
erm
ine
wh
eth
er B�
C�|| D�
E�.
3.A
D�
15,D
B�
12,A
E�
10,a
nd
EC
�8
yes
4.B
D�
9,B
A�
27,a
nd
CE
is o
ne
thir
d of
EA
no
5.A
E�
30,A
C�
45,a
nd
AD
is t
wic
e D
B
yes
CO
OR
DIN
ATE
GEO
MET
RYF
or E
xerc
ises
6–8
,use
th
e fo
llow
ing
info
rmat
ion
.T
rian
gle
AB
Ch
as v
erti
ces
A(�
5,2)
,B(1
,8),
and
C(4
,2).
Poi
nt
Dis
th
e m
idpo
int
of A �
B�an
d E
is t
he
mid
poin
t of
A�C�
.
6.Id
enti
fy t
he
coor
din
ates
of
Dan
d E
.
D(�
2,5)
;E
(�0.
5,2)
7.S
how
th
at B�
C�is
par
alle
l to
D�E�
.
Sin
ce t
he
slo
pe
of
B�C�
is �
2 an
d t
he
slo
pe
of
D�E�
is �
2,B�
C�is
par
alle
l to
D�E�.
8.S
how
th
at D
E�
�1 2� BC
.
Th
e le
ng
th o
f D�
E�is
�11
.25
�o
r an
d t
he
len
gth
of
B�C�
is �
45�.
9.F
ind
xan
d y.
10.F
ind
xan
d y.
x�
6,y
�6.
5x
�2,
y�
4
3 – 2x �
2
x �
3
2 y �
1
3y �
52x
� 1
3 y �
8y
� 5
x �
7
�45�
�2
x
y
O
AC
B
D
E
CB
ED
A
R
UV
T
S
HL
KJ
I
©G
lenc
oe/M
cGra
w-H
ill31
6G
lenc
oe G
eom
etry
1.If
AD
�24
,DB
�27
,an
d E
B�
18,
2.F
ind
x,Q
T,a
nd
TR
if Q
T�
x�
6,fi
nd
CE
.S
R�
12,P
S�
27,a
nd
TR
�x
�4.
1612
;18
;8
Det
erm
ine
wh
eth
er J�
K�|| N�
M�.
3.J
N�
18,J
L�
30,K
M�
21,a
nd
ML
�35
no
4.K
M�
24,K
L�
44,a
nd
NL
��5 6� J
N
yes
CO
OR
DIN
ATE
GEO
MET
RYF
or E
xerc
ises
5 a
nd
6,u
se t
he
foll
owin
g in
form
atio
n.
Tri
angl
e E
FG
has
ver
tice
s E
(�4,
�1)
,F(2
,5),
and
G(2
,�1)
.Poi
nt
Kis
th
e m
idpo
int
of E �
G�an
d H
is t
he
mid
poin
t of
F�G�
.
5.S
how
th
at E�
F�is
par
alle
l to
K�H�
.
Th
e sl
op
e o
f E�
F�is
1 a
nd
th
e sl
op
e o
f K�
H�is
1,
so E�
F�is
par
alle
l to
K�H�
.
6.S
how
th
at K
H�
�1 2� EF
.
Th
e le
ng
th o
f K�
H�is
3�
2�,an
d t
he
len
gth
of
E�F�
is 6
�2�.
7.F
ind
xan
d y.
8.F
ind
xan
d y.
x�
4,y
�9
x�
�5 2� ,y
��9 4�
9.M
APS
Th
e di
stan
ce f
rom
Wil
min
gton
to
Ash
Gro
ve a
lon
g K
enda
ll i
s 82
0 fe
et a
nd
alon
g M
agn
olia
,660
fee
t.If
th
e di
stan
ce b
etw
een
Bee
ch a
nd
Ash
Gro
ve a
lon
g M
agn
olia
is
280
feet
,wh
at i
s th
e di
stan
ce b
etw
een
th
e tw
o st
reet
s al
ong
Ken
dall
?
abo
ut
348
ft
Ash
Grov
e
Beec
h
Mag
nolia
Kend
all
Wilm
ingt
on
4 – 3y �
2
3 x �
4x
� 1
4 y �
43x
� 4
2 – 3y �
31 – 3y
� 6
5 – 4x �
3
x
y
O
EGF H
K
NL
J
KM
S
Q
TR
P
D
EC BA
Pra
ctic
e (
Ave
rag
e)
Par
alle
l Lin
es a
nd
Pro
po
rtio
nal
Par
ts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-4
6-4
Answers (Lesson 6-4)
© Glencoe/McGraw-Hill A13 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csP
aral
lel L
ines
an
d P
rop
ort
ion
al P
arts
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-4
6-4
©G
lenc
oe/M
cGra
w-H
ill31
7G
lenc
oe G
eom
etry
Lesson 6-4
Pre-
Act
ivit
yH
ow d
o ci
ty p
lan
ner
s u
se g
eom
etry
?
Rea
d th
e in
trod
uct
ion
to
Les
son
6-4
at
the
top
of p
age
307
in y
our
text
book
.
Use
a g
eom
etri
c id
ea t
o ex
plai
n w
hy
the
dist
ance
bet
wee
n C
hic
ago
Ave
nu
ean
d O
nta
rio
Str
eet
is s
hor
ter
alon
g M
ich
igan
Ave
nu
e th
an a
lon
g L
ake
Sh
ore
Dri
ve.S
amp
le a
nsw
er:T
he
sho
rtes
t d
ista
nce
bet
wee
n t
wo
par
alle
l lin
es is
th
e p
erp
end
icu
lar
dis
tan
ce.
Rea
din
g t
he
Less
on
1.P
rovi
de t
he
mis
sin
g w
ords
to
com
plet
e th
e st
atem
ent
of e
ach
th
eore
m.T
hen
sta
te t
he
nam
e of
th
e th
eore
m.
a.If
a l
ine
inte
rsec
ts t
wo
side
s of
a t
rian
gle
and
sepa
rate
s th
e si
des
into
cor
resp
ondi
ng
segm
ents
of
len
gth
s,th
en t
he
lin
e is
to
th
e
thir
d si
de.
Co
nver
se o
f th
e Tr
ian
gle
Pro
po
rtio
nal
ity
Th
eore
mb
.A
mid
segm
ent
of a
tri
angl
e is
to
on
e si
de o
f th
e tr
ian
gle
and
its
leng
th is
th
e le
ngth
of
that
sid
e.Tr
ian
gle
Mid
seg
men
t Th
eore
mc.
If a
line
is
to o
ne s
ide
of a
tri
angl
e an
d in
ters
ects
the
oth
er t
wo
side
s
in
dist
inct
poi
nts
,th
en i
t se
para
tes
thes
e si
des
into
of p
ropo
rtio
nal
len
gth
.Tr
ian
gle
Pro
po
rtio
nal
ity
Th
eore
m
2.R
efer
to
the
figu
re a
t th
e ri
ght.
a.N
ame
the
thre
e m
idse
gmen
ts o
f �
RS
T.
U�V�
,V�W�
,U�W�
b.
If R
S�
8,R
U�
3,an
d T
W�
5,fi
nd
the
len
gth
of
each
of
the
mid
segm
ents
.U
V�
5,V
W�
3,U
W�
4c.
Wh
at i
s th
e pe
rim
eter
of
�R
ST
?24
d.
Wh
at i
s th
e pe
rim
eter
of
�U
VW
?12
e.W
hat
are
th
e pe
rim
eter
s of
�R
UV
,�S
VW
,an
d �
TU
W?
12;
12;
12f.
How
are
th
e pe
rim
eter
s of
eac
h o
f th
e fo
ur
smal
l tr
ian
gles
rel
ated
to
the
peri
met
er o
fth
e la
rge
tria
ngl
e?T
he
per
imet
er o
f ea
ch s
mal
l tri
ang
le is
on
e-h
alf
the
per
imet
er o
f th
e la
rge
tria
ng
le.
g.W
ould
th
e re
lati
onsh
ip t
hat
you
fou
nd
in p
art
f ap
ply
to a
ny
tria
ngl
e in
wh
ich
th
em
idpo
ints
of
the
thre
e si
des
are
con
nec
ted?
yes
Hel
pin
g Y
ou
Rem
emb
er
3.A
goo
d w
ay t
o re
mem
ber
a n
ew m
ath
emat
ical
ter
m i
s to
rel
ate
it t
o ot
her
mat
hem
atic
alvo
cabu
lary
th
at y
ou a
lrea
dy k
now
.Wh
at i
s an
eas
y w
ay t
o re
mem
ber
the
defi
nit
ion
of
mid
segm
ent
usi
ng
oth
er g
eom
etri
c te
rms?
Sam
ple
an
swer
:A
mid
seg
men
tis
ase
gm
ent
that
co
nn
ects
th
e m
idp
oin
tso
f tw
o s
ides
of
a tr
ian
gle
.
R U
V W
T
S
seg
men
tstw
op
aral
lel
on
e-h
alf
par
alle
l
par
alle
lp
rop
ort
ion
al
©G
lenc
oe/M
cGra
w-H
ill31
8G
lenc
oe G
eom
etry
Par
alle
l Lin
es a
nd
Co
ng
ruen
t P
arts
Th
ere
is a
th
eore
m s
tati
ng
that
if
thre
e pa
rall
el l
ines
cu
t of
f co
ngr
uen
t se
gmen
ts o
n o
ne
tran
sver
sal,
then
th
ey c
ut
off
con
gru
ent
segm
ents
on
an
y tr
ansv
ersa
l.T
his
can
be
show
n f
oran
y n
um
ber
of p
aral
lel
lin
es.T
he
foll
owin
g dr
afti
ng
tech
niq
ue
use
s th
is f
act
to d
ivid
e a
segm
ent
into
con
gru
ent
part
s.
A �B�
is t
o be
sep
arat
ed i
nto
fiv
e co
ngr
uen
t pa
rts.
Th
is c
an b
e do
ne
very
acc
ura
tely
wit
hou
t u
sin
g a
rule
r.A
ll t
hat
is
nee
ded
is a
com
pass
an
d a
piec
e of
not
eboo
k pa
per.
Ste
p 1
Hol
d th
e co
rner
of
a pi
ece
of n
oteb
ook
pape
r at
po
int
A.
Ste
p 2
Fro
m p
oin
t A
,dra
w a
seg
men
t al
ong
the
pape
r th
at i
s fi
ve s
pace
s lo
ng.
Mar
k w
her
e th
e li
nes
of
th
e n
oteb
ook
pape
r m
eet
the
segm
ent.
Lab
el
the
fift
h p
oin
t,P.
Ste
p 3
Dra
w P�
B�.T
hro
ugh
eac
h o
f th
e ot
her
mar
ks
on A �
P�,c
onst
ruct
a l
ine
para
llel
to
B�P�
.Th
e po
ints
wh
ere
thes
e li
nes
in
ters
ect
A �B�
wil
l di
vide
A �B�
into
fiv
e co
ngr
uen
t se
gmen
ts.
Use
a c
omp
ass
and
a p
iece
of
not
eboo
k p
aper
to
div
ide
each
seg
men
t in
to t
he
give
n n
um
ber
of
con
gru
ent
par
ts.
See
stu
den
t’s w
ork
.
1.si
x co
ngr
uen
t pa
rts
2.se
ven
con
gru
ent
part
sA
B
AB
A
P
B
AB
P
AB
AB
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-4
6-4
Answers (Lesson 6-4)
© Glencoe/McGraw-Hill A14 Glencoe Geometry
Stu
dy G
uid
e a
nd I
nte
rven
tion
Par
ts o
f S
imila
r Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-5
6-5
©G
lenc
oe/M
cGra
w-H
ill31
9G
lenc
oe G
eom
etry
Lesson 6-5
Peri
met
ers
If t
wo
tria
ngl
es a
re s
imil
ar,t
hei
r pe
rim
eter
s h
ave
the
sam
e pr
opor
tion
as
the
corr
espo
ndi
ng
side
s.
If �
AB
C�
�R
ST
,th
en
�A RB S
� �B S
TC��
RATC
��
�A RB S�
��B S
TC ��
� RAC T�
.
Use
th
e d
iagr
am a
bov
e w
ith
�A
BC
��
RS
T.I
f A
B�
24 a
nd
RS
�15
,fi
nd
th
e ra
tio
of t
hei
r p
erim
eter
s.S
ince
�A
BC
��
RS
T,t
he
rati
o of
th
e pe
rim
eter
s of
�A
BC
and
�R
ST
is t
he
sam
e as
th
era
tio
of c
orre
spon
din
g si
des.
Th
eref
ore
��2 14 5�
��8 5�
Eac
h p
air
of t
rian
gles
is
sim
ilar
.Fin
d t
he
per
imet
er o
f th
e in
dic
ated
tri
angl
e.
1.�
XY
Z2.
�B
DE
3345
3.�
AB
C4.
�X
YZ
8455
.4
5.�
AB
C6.
�R
ST
5411
0
25 20
10
PT
SM
R
95
13
12
AD
E
C
B
56
8.7
13
10
XY
ZT
RS
18
202515
17 8
NB
AC
PM
36
9 11 CDE
A
B
44
24
2022
10
TR
XZ
YS
peri
met
er o
f �
AB
C�
��
peri
met
er o
f �
RS
T
AC
BS
RT
Exam
ple
Exam
ple
Exer
cises
Exer
cises
©G
lenc
oe/M
cGra
w-H
ill32
0G
lenc
oe G
eom
etry
Spec
ial S
egm
ents
of
Sim
ilar
Tria
ng
les
Wh
en t
wo
tria
ngl
es a
re s
imil
ar,
corr
espo
ndi
ng
alti
tude
s,an
gle
bise
ctor
s,an
d m
edia
ns
are
prop
orti
onal
to
the
corr
espo
ndi
ng
side
s.A
lso,
in a
ny
tria
ngl
e an
an
gle
bise
ctor
sep
arat
es t
he
oppo
site
sid
e in
to s
egm
ents
th
ath
ave
the
sam
e ra
tio
as t
he
oth
er t
wo
side
s of
th
e tr
ian
gle.
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Par
ts o
f S
imila
r Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-5
6-5
In t
he
figu
re,
�A
BC
��
XY
Z,w
ith
an
gle
bis
ecto
rs a
ssh
own
.Fin
d x
.
Sin
ce �
AB
C�
�X
YZ
,th
e m
easu
res
of t
he
angl
e bi
sect
ors
are
prop
orti
onal
to
the
mea
sure
s of
a p
air
of c
orre
spon
din
g si
des.
�A XB Y�
�� YB
WD �
�2 x4 ��
�1 80 �
10x
�24
(8)
10x
�19
2x
�19
.2
2410 D
W
Y
ZX
C
B
A
8x
S�U�
bis
ects
�R
ST
.Fin
d x
.
Sin
ce S �
U�is
an
an
gle
bise
ctor
,�R T
UU ��
�R TSS �
.
� 2x 0��
�1 35 0�
30x
�20
(15)
30x
�30
0x
�10
x2030
15
UT
S
R
Exam
ple1
Exam
ple1
Exam
ple2
Exam
ple2
Exer
cises
Exer
cises
Fin
d x
for
each
pai
r of
sim
ilar
tri
angl
es.
1.2.
108
3.4.
2.25
8.75
5.6.
12�5 6�
15
2x �
2x
� 1
2816
17
11
x �
7
x
10
10x
8
87
3
3
4x
69
12x
x
36
1820
Answers (Lesson 6-5)
© Glencoe/McGraw-Hill A15 Glencoe Geometry
An
swer
s
Skil
ls P
ract
ice
Par
ts o
f S
imila
r Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-5
6-5
©G
lenc
oe/M
cGra
w-H
ill32
1G
lenc
oe G
eom
etry
Lesson 6-5
Fin
d t
he
per
imet
er o
f th
e gi
ven
tri
angl
e.
1.�
JK
L,i
f �
JK
L�
�R
ST
,RS
�14
,2.
�D
EF
,if
�D
EF
��
AB
C,A
B�
27,
ST
�12
,TR
�10
,an
d L
J�
14B
C�
16,C
A�
25,a
nd
FD
�15
50.4
40.8
3.�
PQ
R,i
f �
PQ
R�
�L
MN
,LM
�16
,4.
�K
LM
,if
�K
LM
��
FG
H,F
G�
30,
MN
�14
,NL
�27
,an
d R
P�
18G
H�
38,H
F�
38,a
nd
KL
�24
3884
.8
Use
th
e gi
ven
in
form
atio
n t
o fi
nd
eac
h m
easu
re.
5.F
ind
FG
if �
RS
T�
�E
FG
,S�H�
is a
n
6.F
ind
MN
if �
AB
C�
�M
NP,
A�D�
is a
n
alti
tude
of
�R
ST
,F �J�
is a
n a
ltit
ude
of
alti
tude
of
�A
BC
,M�Q�
is a
n a
ltit
ude
of
�E
FG
,ST
�6,
SH
�5,
and
FJ
�7.
�M
NP
,AB
�24
,AD
�14
,and
MQ
�10
.5.
8.4
18
Fin
d x
.
7.�
HK
L�
�X
YZ
8.
8�1 3�
8.4
LH
K
18
10
Y
15
x
ZX
7
20
24
x
C
A
BD
PN
QM
JG
E
F
HT
R
S
GL K
M
F
H
LM
PQ
RN
A
BC
D
EF
RS
T
JK
L
©G
lenc
oe/M
cGra
w-H
ill32
2G
lenc
oe G
eom
etry
Fin
d t
he
per
imet
er o
f th
e gi
ven
tri
angl
e.
1.�
DE
F,i
f �
AB
C�
�D
EF
,AB
�36
,2.
�S
TU
,if
�S
TU
��
KL
M,K
L�
12,
BC
�20
,CA
�40
,an
d D
E�
35L
M�
31,M
K�
32,a
nd
US
�28
93.3�
65.6
25
Use
th
e gi
ven
in
form
atio
n t
o fi
nd
eac
h m
easu
re.
3.F
ind
PR
if �
JK
L�
�N
PR
,K�M�
is a
n
4.F
ind
ZY
if �
ST
U�
�X
YZ
,U�A�
is a
n
alti
tude
of
�J
KL
,P �T�
is a
n a
ltit
ude
of
alti
tude
of
�S
TU
,Z�B�
is a
n a
ltit
ude
of
�N
PR
,KL
�28
,KM
�18
,an
d �
XY
Z,U
T�
8.5,
UA
�6,
and
PT
�15
.75.
ZB
�11
.4.
24.5
16.1
5
Fin
d x
.
5.6.
13.5
16.8
PHO
TOG
RA
PHY
For
Exe
rcis
es 7
an
d 8
,use
th
e fo
llow
ing
info
rmat
ion
.F
ran
cin
e h
as a
cam
era
in w
hic
h t
he
dist
ance
fro
m t
he
len
s to
th
e fi
lm i
s 24
mil
lim
eter
s.
7.If
Fra
nci
ne
take
s a
full
-len
gth
ph
otog
raph
of
her
fri
end
from
a d
ista
nce
of
3 m
eter
s an
dth
e h
eigh
t of
her
fri
end
is 1
40 c
enti
met
ers,
wh
at w
ill
be t
he
hei
ght
of t
he
imag
e on
th
efi
lm?
(Hin
t:C
onve
rt t
o th
e sa
me
un
it o
f m
easu
re.)
11.2
mm
8.S
upp
ose
the
hei
ght
of t
he
imag
e on
th
e fi
lm o
f h
er f
rien
d is
15
mil
lim
eter
s.If
Fra
nci
ne
took
a f
ull
-len
gth
sh
ot,w
hat
was
th
e di
stan
ce b
etw
een
th
e ca
mer
a an
d h
er f
rien
d?2.
24 m
x28
2030
402x
� 1
25x
� 4
YX
BZ
TS
AU
MJ
L
K
TN
R
P
KS
LT
UM
B E
CF
DA
Pra
ctic
e (
Ave
rag
e)
Par
ts o
f S
imila
r Tri
ang
les
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-5
6-5
Answers (Lesson 6-5)
© Glencoe/McGraw-Hill A16 Glencoe Geometry
Readin
g t
o L
earn
Math
em
ati
csP
arts
of
Sim
ilar T
rian
gle
s
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-5
6-5
©G
lenc
oe/M
cGra
w-H
ill32
3G
lenc
oe G
eom
etry
Lesson 6-5
Pre-
Act
ivit
yH
ow i
s ge
omet
ry r
elat
ed t
o p
hot
ogra
ph
y?
Rea
d th
e in
trod
uct
ion
to
Les
son
6-5
at
the
top
of p
age
316
in y
our
text
book
.
•H
ow i
s si
mil
arit
y in
volv
ed i
n t
he
proc
ess
of m
akin
g a
phot
ogra
phic
pri
nt
from
a n
egat
ive?
Sam
ple
an
swer
:Th
e p
rin
t is
an
en
larg
ed v
ersi
on
of
the
neg
ativ
e.T
he
imag
e o
n t
he
pri
nt
is s
imila
r to
th
e im
age
on
the
neg
ativ
e,so
wh
en y
ou
loo
k at
th
e p
rin
t,yo
u w
ill s
eeex
actl
y th
e sa
me
thin
gs
in t
he
sam
e p
rop
ort
ion
s as
on
th
en
egat
ive,
just
larg
er.
•W
hy
do p
hot
ogra
pher
s pl
ace
thei
r ca
mer
as o
n t
ripo
ds?
Sam
ple
an
swer
:Th
ree
po
ints
det
erm
ine
a p
lan
e,so
th
e th
ree
feet
of
the
trip
od
are
co
pla
nar
an
d t
he
trip
od
will
no
t w
ob
ble
.
Rea
din
g t
he
Less
on
1.In
th
e fi
gure
,�R
ST
��
UV
W.C
ompl
ete
each
pr
opor
tion
in
volv
ing
the
len
gth
s of
seg
men
ts i
nth
is f
igu
re b
y re
plac
ing
the
ques
tion
mar
k.T
hen
iden
tify
th
e de
fin
itio
n o
r th
eore
m f
rom
th
e li
stbe
low
th
at t
he
com
plet
ed p
ropo
rtio
n i
llu
stra
tes.
i.D
efin
itio
n o
f co
ngr
uen
t po
lygo
ns
ii.
Def
init
ion
of
sim
ilar
pol
ygon
s
iii.
Pro
port
ion
al P
erim
eter
s T
heo
rem
iv.
An
gle
Bis
ecto
rs T
heo
rem
v.S
imil
ar t
rian
gles
hav
e co
rres
pon
din
g al
titu
des
prop
orti
onal
to
corr
espo
ndi
ng
side
s.
vi.
Sim
ilar
tri
angl
es h
ave
corr
espo
ndi
ng
med
ian
s pr
opor
tion
al t
o co
rres
pon
din
g si
des.
vii.
Sim
ilar
tria
ngle
s ha
ve c
orre
spon
ding
ang
le b
isec
tors
pro
port
iona
l to
corr
espo
ndin
g si
des.
a.�R
S�
S ?T�
TR
��
� URS V�
UV
�V
W�
WU
;iii
b.
� URWT �
��S ?X �
VY
;v
c.�R U
M N��
� V? W�S
T;
vid
.� UR
S V��
�S ?T �V
W;
ii
e.�R P
SP ��
� S? T�R
T;
ivf.
�U ?N ��
�U RW T�
RM
;vi
g.� WT
P Q��
�R ?T �U
W;
vii
h.
�U VWW �
�� Q? V�
UQ
;iv
Hel
pin
g Y
ou
Rem
emb
er
2.A
goo
d w
ay t
o re
mem
ber
a la
rge
amou
nt
of i
nfo
rmat
ion
is
to r
emem
ber
key
wor
ds.W
hat
key
wor
ds w
ill
hel
p yo
u r
emem
ber
the
feat
ure
s of
sim
ilar
tri
angl
es t
hat
are
pro
port
ion
alto
th
e le
ngt
hs
of t
he
corr
espo
ndi
ng
side
s?
per
imet
ers,
alti
tud
es,a
ng
le b
isec
tors
,med
ian
s
W
QN
YU
V
T
PM
XR
S
©G
lenc
oe/M
cGra
w-H
ill32
4G
lenc
oe G
eom
etry
Pro
po
rtio
ns
for
Sim
ilar T
rian
gle
sR
ecal
l th
at i
f a
lin
e cr
osse
s tw
o si
des
of a
tri
angl
e an
d is
par
alle
l to
th
e th
ird
side
,th
en t
he
lin
e se
para
tes
the
two
side
s th
at i
t cr
osse
s in
to s
egm
ents
of
prop
orti
onal
len
gth
s.
You
can
wri
te m
any
prop
orti
ons
by i
den
tify
ing
sim
ilar
tri
angl
es i
n t
he
foll
owin
g di
agra
m.I
n t
he
diag
ram
,AM
� ��
�B
N� �
� ,A
E��
��
FL
� ��
�M
R� �
� ,an
d D
Q��
��
ER
� �� .
An
swer
eac
h q
ues
tion
.Use
th
e d
iagr
am a
bov
e.
1.N
ame
a tr
ian
gle
sim
ilar
to
�G
NP
.2.
Nam
e a
tria
ngl
e si
mil
ar t
o �
CJ
H.
�A
MP
or
�G
BA
or
�A
FG
�C
RP
3.N
ame
two
tria
ngl
es s
imil
ar t
o �
JK
S.
4.N
ame
a tr
ian
gle
sim
ilar
to
�A
CP
.
Sam
ple
an
swer
:�
JLR
,�C
DS
�G
HP
Com
ple
te e
ach
pro
por
tion
.
5.�A A
G P��
�A ?F �A
M6.
� CCHP �
��C ?R �
CJ
7.� JJ RS �
�� J? L�
JK
8.�P P
H C��
�P ?G �PA
9.�E L
RR ��
� J? R�C
R10
.�M M
N P��
� A? P�G
A
Sol
ve.
11.I
f C
J�
16,J
R�
48,a
nd
LR
�30
,fin
d E
L.
10
12.I
f D
K�
5,K
S�
7,an
d C
J�
8,fi
nd
JS
.11
.2
13.I
f M
N�
12,N
P�
32,a
nd
AP
�48
,fin
d A
G.R
oun
d to
th
e n
eare
stte
nth
.13
.1
14.I
f C
H�
18,H
P�
82,a
nd
CR
�13
0,fi
nd
CJ
.23
.4
15.W
rite
th
ree
mor
e pr
oble
ms
that
can
be
solv
ed u
sin
g th
e di
agra
m a
bove
.S
ee s
tud
ents
’wo
rk.
AB
CD
E
FG
HJ
KL
MN
PR
Q
S
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-5
6-5
Answers (Lesson 6-5)
© Glencoe/McGraw-Hill A17 Glencoe Geometry
An
swer
s
Stu
dy G
uid
e a
nd I
nte
rven
tion
Fra
ctal
s an
d S
elf-
Sim
ilari
ty
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-6
6-6
©G
lenc
oe/M
cGra
w-H
ill32
5G
lenc
oe G
eom
etry
Lesson 6-6
Ch
arac
teri
stic
s o
f Fr
acta
lsT
he
act
of r
epea
tin
g a
proc
ess
over
an
d ov
er,s
uch
as
fin
din
g a
thir
d of
a s
egm
ent,
then
a t
hir
d of
th
e n
ew s
egm
ent,
and
so o
n,i
s ca
lled
ite
rati
on.
Wh
en t
he
proc
ess
of i
tera
tion
is
appl
ied
to s
ome
geom
etri
c fi
gure
s,th
e re
sult
s ar
e ca
lled
frac
tals
.For
obj
ects
su
ch a
s fr
acta
ls,w
hen
a p
orti
on o
f th
e ob
ject
has
th
e sa
me
shap
e or
char
acte
rist
ics
as t
he
enti
re o
bjec
t,th
e ob
ject
can
be
call
ed s
elf-
sim
ilar
.
In t
he
dia
gram
at
the
righ
t,n
otic
e th
at t
he
det
ails
at
each
sta
ge a
re s
imil
ar t
o th
e d
etai
ls a
t S
tage
1.
1.F
ollo
w t
he
iter
atio
n p
roce
ss b
elow
to
prod
uce
a f
ract
al.
Sta
ge 1
• D
raw
a s
quar
e.•
Dra
w a
n i
sosc
eles
rig
ht
tria
ngl
e on
th
e to
p si
de o
f th
e sq
uar
e.U
se t
he
side
of
the
squ
are
as t
he
hyp
oten
use
of
the
tria
ngl
e.•
Dra
w a
squ
are
on e
ach
leg
of
the
righ
t tr
ian
gle.
Sta
ge 2
Rep
eat
the
step
s in
Sta
ge 1
,dra
win
g an
iso
scel
es
tria
ngl
e an
d tw
o sm
all
squ
ares
for
eac
h o
f th
e sm
all
squ
ares
fro
m S
tage
1.
Sta
ge 3
Rep
eat
the
step
s in
Sta
ge 1
for
eac
h o
f th
e sm
alle
st s
quar
es i
n S
tage
2.
2.Is
th
e fi
gure
pro
duce
d in
Sta
ge 3
sel
f-si
mil
ar?
yes
Stag
e 1
Stag
e 2
Stag
e 3
Exer
cises
Exer
cises
Exam
ple
Exam
ple
©G
lenc
oe/M
cGra
w-H
ill32
6G
lenc
oe G
eom
etry
No
ng
eom
etri
c It
erat
ion
An
ite
rati
ve p
roce
ss c
an b
e ap
plie
d to
an
alg
ebra
icex
pres
sion
or
equ
atio
n.T
he
resu
lt i
s ca
lled
a r
ecu
rsiv
e fo
rmu
la.
Fin
d t
he
valu
e of
x3 ,
wh
ere
the
init
ial
valu
e of
xis
2.R
epea
t th
ep
roce
ss t
hre
e ti
mes
an
d d
escr
ibe
the
pat
tern
.
Init
ial
valu
e:2
Fir
st t
ime:
23�
8S
econ
d ti
me:
83�
512
Th
ird
tim
e:51
23�
134,
217,
728
Th
e re
sult
of
each
ste
p of
th
e it
erat
ion
is
use
d fo
r th
e n
ext
step
.For
th
is e
xam
ple,
the
xva
lues
are
gre
ater
wit
h e
ach
ite
rati
on.T
her
e is
no
max
imu
m v
alu
e,so
th
e va
lues
are
desc
ribe
d as
app
roac
hin
g in
fin
ity.
For
Exe
rcis
es 1
–5,f
ind
th
e va
lue
of e
ach
exp
ress
ion
.Th
en u
se t
hat
val
ue
as t
he
nex
t x
in t
he
exp
ress
ion
.Rep
eat
the
pro
cess
th
ree
tim
es,a
nd
des
crib
e yo
ur
obse
rvat
ion
s.
1.�
x�,w
her
e x
init
iall
y eq
ual
s 5
�5�
�2.
24;
1.50
,1.2
2,1.
11;
the
xva
lues
get
sm
alle
r w
ith
eac
h it
erat
ion
.
2.�1 x� ,
wh
ere
xin
itia
lly
equ
als
2
�1 2� ;2,
�1 2� ,2;
the
xva
lues
alt
ern
ate
bet
wee
n �1 2�
and
2.
3.3x
,wh
ere
xin
itia
lly
equ
als
1
31�
3;33
�27
,327
�7.
6 �
1012
;th
e x
valu
es g
et g
reat
er w
ith
eac
hit
erat
ion
,ap
pro
ach
ing
infi
nit
y.
4.x
�5
wh
ere
xin
itia
lly
equ
als
10
10 �
5 �
5;0,
�5,
�10
;th
e x
valu
es d
ecre
ase
by 5
wit
h e
ach
iter
atio
n.
5.x2
�4,
wh
ere
xin
itia
lly
equ
als
16
162
�4
�25
2;25
22�
4 �
63,5
00;
63,5
002
�4
�4.
0 �
109 ;
(4.0
�10
9 )2
�4
�1.
6 �
1019
;th
e x
valu
es in
crea
se w
ith
eac
h it
erat
ion
,ap
pro
ach
ing
infi
nit
y.
6.H
arpe
sh p
aid
$100
0 fo
r a
savi
ngs
cer
tifi
cate
.It
earn
s in
tere
st a
t an
an
nu
al r
ate
of 2
.8%
,an
d in
tere
st i
s ad
ded
to t
he
cert
ific
ate
each
yea
r.W
hat
wil
l th
e ce
rtif
icat
e be
wor
th a
fter
fou
r ye
ars?
$111
6.79
Stu
dy G
uid
e a
nd I
nte
rven
tion
(con
tinued
)
Fra
ctal
s an
d S
elf-
Sim
ilari
ty
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-6
6-6
Exer
cises
Exer
cises
Exam
ple
Exam
ple
Answers (Lesson 6-6)
© Glencoe/McGraw-Hill A18 Glencoe Geometry
Skil
ls P
ract
ice
Fra
ctal
s an
d S
elf-
Sim
ilari
ty
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-6
6-6
©G
lenc
oe/M
cGra
w-H
ill32
7G
lenc
oe G
eom
etry
Lesson 6-6
Sta
ges
1 an
d 2
of
a fr
acta
l k
now
n a
s th
e C
anto
r se
t ar
e sh
own
.To
get
Sta
ge 2
,th
ese
gmen
t in
Sta
ge 1
is
tris
ecte
d,a
nd
th
e in
teri
or o
f th
e m
idd
le s
egm
ent
is r
emov
ed.
(Th
e in
teri
or o
f a
segm
ent
is t
he
segm
ent
wit
h i
ts e
nd
poi
nts
rem
oved
.) T
he
pro
cess
is
rep
eate
d f
or s
ub
seq
uen
t st
ages
.
1.D
raw
sta
ges
3 an
d 4
of t
he
Can
tor
set.
2.H
ow m
any
segm
ents
are
th
ere
in S
tage
3?
Sta
ge 4
?
4 se
gm
ents
;8
seg
men
ts
3.W
hat
hap
pen
s to
th
e le
ngt
h o
f th
e li
ne
segm
ents
in
eac
h s
tage
?
Th
e lin
e se
gm
ents
dec
reas
e in
len
gth
by
two
-th
ird
s at
eac
h s
tag
e.
4.T
he
Can
tor
set
is t
he
set
of p
oin
ts a
fter
in
fin
itel
y m
any
iter
atio
ns.
Is t
he
Can
tor
set
self
-sim
ilar
?
yes
Fin
d t
he
valu
e of
eac
h e
xpre
ssio
n.T
hen
,use
th
at v
alu
e as
th
e n
ext
xin
th
eex
pre
ssio
n.R
epea
t th
e p
roce
ss t
hre
e ti
mes
,an
d d
escr
ibe
you
r ob
serv
atio
ns.
5.2x
�3,
wh
ere
xin
itia
lly
equ
als
5
2(5)
�3
�7;
2(7)
�3
�11
;2(
11)
�3
�19
;2(
19)
�3
�35
Th
e it
erat
es in
crea
se b
y co
nse
cuti
ve p
ow
ers
of
2.
6.� 2x � ,
wh
ere
xin
itia
lly
equ
als
1
�1 2��
0.5;
�0 2.5 ��
0.25
;�0.
225 ��
0.12
5;�0.
1 225 ��
0.06
25
Th
e it
erat
es a
re h
alve
d w
ith
eac
h it
erat
ion
,ap
pro
ach
ing
zer
o.
Fin
d t
he
firs
t th
ree
iter
ates
of
each
exp
ress
ion
.
7.x2
�1,
wh
ere
xin
itia
lly
equ
als
28.
�1 x� ,w
her
e x
init
iall
y eq
ual
s 8
3,8,
630.
125,
8,0.
125
Stag
e 3
Stag
e 4
Stag
e 1
Stag
e 2
©G
lenc
oe/M
cGra
w-H
ill32
8G
lenc
oe G
eom
etry
An
art
ist
is d
esig
nin
g a
boo
k c
over
an
d w
ants
to
show
a c
opy
of t
he
boo
k c
over
in
the
low
er l
eft
corn
er o
f th
e co
ver.
Aft
er S
tage
s 1
and
2 o
f th
e d
esig
n,h
e re
aliz
esth
at t
he
des
ign
is
dev
elop
ing
into
a f
ract
al!
1.D
raw
Sta
ges
3 an
d 4
of t
he
book
cov
er f
ract
al.
2.A
fter
in
fin
itel
y m
any
iter
atio
ns,
wil
l th
e re
sult
be
a se
lf s
imil
ar f
ract
al?
yes
3.O
n w
hat
par
t of
th
e bo
ok c
over
sh
ould
you
foc
us
you
r at
ten
tion
to
be s
ure
you
can
fin
d a
copy
of
the
enti
re f
igu
re?
the
low
er le
ft c
orn
er
Fin
d t
he
valu
e of
eac
h e
xpre
ssio
n.T
hen
,use
th
at v
alu
e as
th
e n
ext
xin
th
eex
pre
ssio
n.R
epea
t th
e p
roce
ss t
hre
e ti
mes
an
d d
escr
ibe
you
r ob
serv
atio
ns.
4.2(
x�
3),w
her
e x
init
iall
y eq
ual
s 0
5.� 2x �
�3,
wh
ere
xin
itia
lly
equ
als
10
6,18
,42,
902,
�2,
�4,
�5
Th
e d
iffe
ren
ce b
etw
een
val
ues
T
he
dif
fere
nce
bet
wee
n v
alu
es is
d
ou
ble
s w
ith
eac
h t
erm
.m
ult
iplie
d b
y �1 2�
wit
h e
ach
ter
m.
Fin
d t
he
firs
t th
ree
iter
ates
of
each
exp
ress
ion
.
6.� 2x �
�1,
wh
ere
xin
itia
lly
equ
als
17.
2x2 ,
wh
ere
xin
itia
lly
equ
als
2
1.5;
1.75
;1.
875
8;12
8;32
,768
8.H
OU
SIN
GT
he
An
drew
s pu
rch
ased
a h
ouse
for
$96
,000
.Th
e re
al e
stat
e ag
ent
wh
o so
ldth
e h
ouse
sai
d th
at c
ompa
rabl
e h
ouse
s in
th
e ar
ea a
ppre
ciat
e at
a r
ate
of 4
.5%
per
yea
r.If
th
is p
atte
rn c
onti
nu
es,w
hat
wil
l be
th
e va
lue
of t
he
hou
se i
n t
hre
e ye
ars?
Rou
nd
toth
e n
eare
st w
hol
e n
um
ber.
$109
,552
Stag
e 3 MAT
HM
ATH
MAT
H
Stag
e 4 MAT
HM
ATH
MAT
HM
ATH
Stag
e 1 MAT
HSt
age
2 MAT
HM
ATH
Pra
ctic
e (
Ave
rag
e)
Fra
ctal
s an
d S
elf-
Sim
ilari
ty
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-6
6-6
Answers (Lesson 6-6)
© Glencoe/McGraw-Hill A19 Glencoe Geometry
An
swer
s
Readin
g t
o L
earn
Math
em
ati
csF
ract
als
and
Sel
f-S
imila
rity
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-6
6-6
©G
lenc
oe/M
cGra
w-H
ill32
9G
lenc
oe G
eom
etry
Lesson 6-6
Pre-
Act
ivit
yH
ow i
s m
ath
emat
ics
fou
nd
in
nat
ure
?
Rea
d th
e in
trod
uct
ion
to
Les
son
6-6
at
the
top
of p
age
325
in y
our
text
book
.
Nam
e tw
o ob
ject
s fr
om n
atu
re o
ther
th
an b
rocc
oli
in w
hic
h a
sm
all
piec
ere
sem
bles
th
e w
hol
e.S
amp
le a
nsw
er:
rin
gs
of
a tr
ee t
run
k;sp
iral
sh
ell
Rea
din
g t
he
Less
on
1.M
atch
eac
h d
efin
itio
n f
rom
th
e fi
rst
colu
mn
wit
h a
ter
m f
rom
th
e se
con
d co
lum
n.(
Som
ew
ords
of
phra
ses
in t
he
seco
nd
colu
mn
may
be
use
d m
ore
than
on
ce o
r n
ot a
t al
l.)
Ph
rase
Ter
ma.
a ge
omet
ric
figu
re t
hat
is
crea
ted
usi
ng
iter
atio
niii
i.si
mil
ar
b.
a pa
tter
n i
n w
hic
h s
mal
ler
and
smal
ler
deta
ils
of a
sh
ape
hav
eii
.ite
rati
on
the
sam
e ge
omet
ric
char
acte
rist
ics
as t
he
orig
inal
sh
ape
ivii
i.fr
acta
lc.
the
resu
lt o
f tr
ansl
atin
g an
ite
rati
ve p
roce
ss i
nto
a f
orm
ula
or
iv.
self
-sim
ilar
alge
brai
c eq
uat
ion
vv.
recu
rsio
nd
.th
e pr
oces
s of
rep
eati
ng
the
sam
e pr
oced
ure
ove
r an
d ov
erfo
rmu
laag
ain
iivi
.con
gru
ent
2.R
efer
to
the
thre
e pa
tter
ns
belo
w.
i. ii.
iii.
a.G
ive
the
spec
ial
nam
e fo
r ea
ch o
f th
e fr
acta
ls y
ou o
btai
n b
y co
nti
nu
ing
the
patt
ern
wit
hou
t en
d.i.
Ko
ch s
no
wfl
ake;
ii.S
ierp
insk
i tri
ang
le;
iii.K
och
cu
rve
b.
Wh
ich
of
thes
e fr
acta
ls a
re s
elf-
sim
ilar
?S
ierp
insk
i tri
ang
le,K
och
cu
rve
c.W
hic
h o
f th
ese
frac
tals
are
str
ictl
y se
lf-s
imil
ar?
Sie
rpin
ski t
rian
gle
Hel
pin
g Y
ou
Rem
emb
er3.
A g
ood
way
to
rem
embe
r a
new
mat
hem
atic
al t
erm
is
to r
elat
e it
to
ever
yday
En
glis
hw
ords
.Th
e w
ord
frac
tal
is r
elat
ed t
o fr
acti
onan
d fr
agm
ent.
Use
you
r di
ctio
nar
y to
fin
dat
lea
st o
ne
defi
nit
ion
for
eac
h o
f th
ese
wor
ds t
hat
you
th
ink
is r
elat
ed t
o th
e m
ean
ing
ofth
e w
ord
frac
tal
and
expl
ain
th
e co
nn
ecti
on.
Sam
ple
an
swer
:F
ract
ion
:a
smal
lp
art;
a d
isco
nn
ecte
d p
iece
;fr
agm
ent:
a p
iece
det
ach
ed o
r b
roke
n o
ff;
som
eth
ing
inco
mp
lete
;in
a f
ract
al,t
he
smal
ler
pie
ces
of
the
shap
ere
sem
ble
th
e w
ho
le.
©G
lenc
oe/M
cGra
w-H
ill33
0G
lenc
oe G
eom
etry
Th
e M
öb
ius
Str
ipA
Möb
ius
stri
p is
a s
peci
al s
urf
ace
wit
h o
nly
on
e si
de.I
t w
as d
isco
vere
d by
Au
gust
Fer
din
and
Möb
ius,
a G
erm
an a
stro
nom
er a
nd
mat
hem
atic
ian
.
1.T
o m
ake
a M
öbiu
s st
rip,
cut
a st
rip
of p
aper
abo
ut
16 i
nch
es l
ong
and
1in
ch w
ide.
Mar
k th
e en
ds w
ith
th
e le
tter
s A
,B,C
,an
d D
as s
how
n b
elow
.
Tw
ist
the
pape
r on
ce,c
onn
ecti
ng
Ato
Dan
d B
to C
.Tap
e th
e en
dsto
geth
er o
n b
oth
sid
es.
See
stu
den
ts’w
ork
.
2.U
se a
cra
yon
or
pen
cil
to s
had
e on
e si
de o
f th
e pa
per.
Sh
ade
arou
nd
the
stri
p u
nti
l yo
u g
et b
ack
to w
her
e yo
u s
tart
ed.W
hat
hap
pen
s?
Th
e en
tire
str
ip is
sh
aded
on
bo
th s
ides
.
3.W
hat
do
you
th
ink
wil
l h
appe
n i
f yo
u c
ut
the
Möb
ius
stri
p do
wn
th
em
iddl
e? T
ry i
t.
Inst
ead
of
two
loo
ps,
you
get
on
e lo
op
th
at is
tw
ice
as lo
ng
as t
he
ori
gin
al o
ne.
4.M
ake
anot
her
Möb
ius
stri
p.S
tart
ing
a th
ird
of t
he
way
in
fro
m o
ne
edge
,cu
t ar
oun
d th
e st
rip,
stay
ing
alw
ays
the
sam
e di
stan
ce i
n f
rom
th
e ed
ge.
Wh
at h
appe
ns?
Two
inte
rlo
ckin
g lo
op
s ar
e fo
rmed
.
5.S
tart
wit
h a
not
her
lon
g st
rip
of p
aper
.Tw
ist
the
pape
r tw
ice
and
con
nec
tth
e en
ds.W
hat
hap
pen
s w
hen
you
cu
t do
wn
th
e ce
nte
r of
th
is s
trip
?
Two
inte
rlo
ckin
g lo
op
s ar
e fo
rmed
.
6.S
tart
wit
h an
othe
r lo
ng s
trip
of
pape
r.T
wis
t th
e pa
per
thre
e ti
mes
and
conn
ect
the
ends
.Wha
t ha
ppen
s w
hen
you
cut
dow
n th
e ce
nter
of
this
str
ip?
On
e tw
o-s
ided
loo
p w
ith
a k
no
t is
fo
rmed
.
A B
C D
En
rich
men
t
NA
ME
____
____
____
____
____
____
____
____
____
____
____
__D
AT
E__
____
____
__P
ER
IOD
____
_
6-6
6-6
Answers (Lesson 6-6)
© Glencoe/McGraw-Hill A20 Glencoe Geometry
Chapter 6 Assessment Answer Key Form 1 Form 2APage 331 Page 332 Page 333
(continued on the next page)
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
A
D
B
B
B
D
B
C
A
B
11.
12.
13.
14.
15.
16.
17.
18.
19.
B:
D
D
A
B
A
B
D
C
A
9.6
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
A
A
D
C
B
C
C
A
C
B
© Glencoe/McGraw-Hill A21 Glencoe Geometry
Chapter 6 Assessment Answer KeyForm 2A (continued) Form 2BPage 334 Page 335 Page 336
An
swer
s
11.
12.
13.
14.
15.
16.
17.
18.
19.
B:
D
D
B
B
C
A
A
A
C
32
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
A
C
B
D
B
B
D
B
B
C
11.
12.
13.
14.
15.
16.
17.
18.
19.
B:
D
A
C
D
B
A
D
C
D
26
© Glencoe/McGraw-Hill A22 Glencoe Geometry
Chapter 6 Assessment Answer KeyForm 2CPage 337 Page 338
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
3:7
Yes; corres. �are �.
30 ft
18
42.5 ft
6.3
18.2
No; the sides arenot proportional.
72
�PQR � �STR
2�12
�
12.
13.
14.
15.
16.
17.
18.
19.
B:
9.5
42
47
2
3
14.4
4
123
7
© Glencoe/McGraw-Hill A23 Glencoe Geometry
Chapter 6 Assessment Answer KeyForm 2DPage 339 Page 340
An
swer
s
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
5:11
No; thecorrespondingsides are notproportional.
48 ft
�43
�
183�13
� ft
5.5
15
yes; SSSSimilarity
75
�XYZ � �XNM
7.5
12.
13.
14.
15.
16.
17.
18.
19.
B:
2.2
96
85
9.5
2
5�58
�
28
472
4.8
© Glencoe/McGraw-Hill A24 Glencoe Geometry
Chapter 6 Assessment Answer KeyForm 3Page 341 Page 342
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
4:3
No; corr. sides arenot proportional.
yes; SAS
30 cm by 7.5 cm
5.85
5
104 in.
82
48
�83
�
5
12.
13.
14.
15.
16.
17.
18.
19.
B:
11�37
�
7.2
�85
�
60
24.8
24 ft
9
10 in.
25.6
Chapter 6 Assessment Answer KeyPage 343, Open-Ended Assessment
Scoring Rubric
© Glencoe/McGraw-Hill A25 Glencoe Geometry
Score General Description Specific Criteria
• Shows thorough understanding of the concepts of ratios,properties of proportions, similar figures, similar triangles,dividing segments into parts, proportional parts of triangles,corresponding perimeters and altitudes, fractals, anditeration.
• Uses appropriate strategies to solve problems.• Computations are correct.• Written explanations are exemplary.• Figures are accurate and appropriate.• Goes beyond requirements of some or all problems.
• Shows an understanding of the concepts of ratios,properties of proportions, similar figures, similar triangles,dividing segments into parts, proportional parts oftriangles, corresponding perimeters and altitudes, fractals,and iteration.
• Uses appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are effective.• Figures are mostly accurate and appropriate.• Satisfies all requirements of problems.
• Shows an understanding of most of the concepts of ratios,properties of proportions, similar figures, similar triangles,dividing segments into parts, proportional parts of triangles,corresponding perimeters and altitudes, fractals, anditeration.
• May not use appropriate strategies to solve problems.• Computations are mostly correct.• Written explanations are satisfactory.• Figures are mostly accurate.• Satisfies the requirements of most of the problems.
• Final computation is correct.• No written explanations or work is shown to substantiate
the final computation.• Figures may be accurate but lack detail or explanation.• Satisfies minimal requirements of some of the problems.
• Shows little or no understanding of most of the concepts ofratios, properties of proportions, similar figures, similartriangles, dividing segments into parts, proportional partsof triangles, corresponding perimeters and altitudes,fractals, and iteration.
• Does not use appropriate strategies to solve problems.• Computations are incorrect.• Written explanations are unsatisfactory.• Figures are inaccurate or inappropriate.• Does not satisfy requirements of problems.• No answer may be given.
0 UnsatisfactoryAn incorrect solutionindicating no mathematicalunderstanding of theconcept or task, or nosolution is given
1 Nearly Unsatisfactory A correct solution with nosupporting evidence orexplanation
2 Nearly SatisfactoryA partially correctinterpretation and/orsolution to the problem
3 SatisfactoryA generally correct solution,but may contain minor flawsin reasoning or computation
4 SuperiorA correct solution that is supported by well-developed, accurateexplanations
An
swer
s
© Glencoe/McGraw-Hill A26 Glencoe Geometry
Chapter 6 Assessment Answer KeyPage 343, Open-Ended Assessment
Sample Answers
In addition to the scoring rubric found on page A25, the following sample answers may be used as guidance in evaluating open-ended assessment items.
1. �53� � �
160� or �
135� � �
120�
2. a. 3:2:2 (Note: Check to be sure that the sum of any two sides of thetriangle is greater than the third side. For example, a ratio of 1:1:2would not be acceptable.)
b. 18, 12, 12
c. 12, 8, 8
3. �ABH � �CDI � �GFI � �ADG � �GDE � �CFE � �AGE by the AAPostulate.
4.
5. a.
Mark the midpoint of each side of �PQR. Connect these points to form�XYZ. Since each segment of �XYZ is a midsegment of �PQR, itslength will be half the corresponding side, and therefore the perimeterwill be half the perimeter of �PQR.
b.
c. Each of the triangles are similar by SSS (since their sides are half thelength of the preceding sides), so the figure is strictly self-similar andthus a fractal.
d. 6
P R
Q
X
Y Z
P R
Q
X
Y Z
FD
CA
B
1
1E
4–3
3–4
© Glencoe/McGraw-Hill A27 Glencoe Geometry
Chapter 6 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 344 Page 345 Page 346
An
swer
s
Quiz 2Page 345
Quiz 4Page 346
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
ratio
scale factor
midsegment
extremes
means
fractal
similar polygons
iteration
proportion
self-similar
Sample answer: Given
a proportion �ba
� � �dc
�,
ad � bc.
Sample answer: Eachpart of the figure, no
matter where it islocated or what size isselected, contains the
same figure as thewhole.
1.
2.
3.
4.
5.
1:30
33 in.
�BAC � �QPR or�ABC � �QPR by
SAS.
x � 4;scale factor 5:4
B
1.
2.
3.
4.
5.
yes; AA
No; the sides arenot proportional.
�ABD � �CDE or�ADB � �CDF;
8
22�12
� ft
6
1.
2.
3.
4.
5.
22�12
�
13�13
�
20
9
14
1.
2.
3.
4.
81
7, 27, 107
6, 30, 870
Stage 3
© Glencoe/McGraw-Hill A28 Glencoe Geometry
Chapter 6 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 347 Page 348
Part I
Part II
6.
7.
8.
9.
10.
No; SAS doesnot apply.yes; SAS
28; �83
�
120
3.6 in.
1.
2.
3.
4.
5.
B
D
C
D
B
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
(�5.5, �8)
11
true
6
perpendicular
y � 4x � 47
yes; TU � WX � �40�,UV � XY � �53�, and
VT � YW � �41�
�NMQ � �NPR;
L�M� � L�P�
x � 0.5, y � 2, z � 1
m�BEG � m�CEG
�H � �H, J�L� || K�M�,and �HJL � �HKM,thus �HJL � �HKM
by AA Similarity.
9
a � 1; b � 2
© Glencoe/McGraw-Hill A29 Glencoe Geometry
An
swer
s
Chapter 6 Assessment Answer KeyStandardized Test Practice
Page 349 Page 350
1.
2.
3.
4.
5.
6.
7.
8. E F G H
A B C D
E F G H
A B C D
E F G H
A B C D
E F G H
A B C D9. 10.
11. 12.
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
0 0 0
.. ./ /
.
99 9 987654321
87654321
87654321
87654321
13.
14.
15.
16.
�3
85.5
11 � x � 53
64.5 cm
1 1 9 2 5
1 2 8 1 6