Chapter 4 Resource Masters - Math Problem Solvingjaeproblemsolving.weebly.com/.../5/1/...chapter_4.pdf · ©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the

Chapter 4Resource 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 4 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: 0-07-860181-9 GeometryChapter 4 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 4-1Study Guide and Intervention . . . . . . . . 183–184Skills Practice . . . . . . . . . . . . . . . . . . . . . . . 185Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 186Reading to Learn Mathematics . . . . . . . . . . 187Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . 188







Chapter 4 AssessmentChapter 4 Test, Form 1 . . . . . . . . . . . . 225–226Chapter 4 Test, Form 2A . . . . . . . . . . . 227–228Chapter 4 Test, Form 2B . . . . . . . . . . . 229–230Chapter 4 Test, Form 2C . . . . . . . . . . . 231–232Chapter 4 Test, Form 2D . . . . . . . . . . . 233–234Chapter 4 Test, Form 3 . . . . . . . . . . . . 235–236Chapter 4 Open-Ended Assessment . . . . . . 237Chapter 4 Vocabulary Test/Review . . . . . . . 238Chapter 4 Quizzes 1 & 2 . . . . . . . . . . . . . . . 239Chapter 4 Quizzes 3 & 4 . . . . . . . . . . . . . . . 240Chapter 4 Mid-Chapter Test . . . . . . . . . . . . 241Chapter 4 Cumulative Review . . . . . . . . . . . 242Chapter 4 Standardized Test Practice . 243–244

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

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

© Glencoe/McGraw-Hill iv Glencoe Geometry

Teacher’s Guide to Using theChapter 4 Resource Masters

The Fast File Chapter Resource system allows you to conveniently file the resourcesyou use most often. The Chapter 4 Resource Masters includes the core materials neededfor Chapter 4. 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 4-1.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 4-1.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 4Resources 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 232–233. 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 _____

44

© 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 4.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

acute triangle

base angles

congruence transformation

kuhn·GROO·uhns

congruent triangles

coordinate proof

corollary

equiangular triangle

equilateral triangle

exterior angle

(continued on the next page)

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

flow proof

included angle

included side

isosceles triangle

obtuse triangle

remote interior angles

right triangle

scalene triangle

SKAY·leen

vertex angle

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

44

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

44

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 4. 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 4.1Angle Sum Theorem

Theorem 4.2Third Angle Theorem

Theorem 4.3Exterior Angle Theorem

Theorem 4.4

Theorem 4.5Angle-Angle-Side Congruence (AAS)

Theorem 4.6Leg-Leg Congruence (LL)

Theorem 4.7Hypotenuse-Angle Congruence (HA)


© Glencoe/McGraw-Hill x Glencoe Geometry

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 4.8Leg-Angle Congruence (LA)

Theorem 4.9Isosceles Triangle Theorem

Theorem 4.10

Postulate 4.1Side-Side-Side Congruence (SSS)

Postulate 4.2Side-Angle-Side Congruence (SAS)

Postulate 4.3Angle-Side-Angle Congruence (ASA)

Postulate 3.4Hypotenuse-Leg Congruence(HL)

Learning to Read MathematicsProof Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

44

Study Guide and InterventionClassifying Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

© Glencoe/McGraw-Hill 183 Glencoe Geometry

Less

on

4-1

Classify Triangles by Angles One way to classify a triangle is by the measures of its angles.

• If one of the angles of a triangle is an obtuse angle, then the triangle is an obtuse triangle.

• If one of the angles of a triangle is a right angle, then the triangle is a right triangle.

• If all three of the angles of a triangle are acute angles, then the triangle is an acute triangle.

• If all three angles of an acute triangle are congruent, then the triangle is an equiangular triangle.

Classify each triangle.

a.

All three angles are congruent, so all three angles have measure 60°.The triangle is an equiangular triangle.

b.

The triangle has one angle that is obtuse. It is an obtuse triangle.

c.

The triangle has one right angle. It is a right triangle.

Classify each triangle as acute, equiangular, obtuse, or right.

1. 2. 3.

4. 5. 6.60�

28� 92�F D

B

45�

45�90�X Y

W

65� 65�

50�

U V

T

60� 60�

60�

Q

R S

120�

30� 30�N O

P

67�

90� 23�

K

L M

90�

60� 30�

G

H J

25�35�

120�

D F

E

60�

A

B C

ExampleExample

ExercisesExercises


Classify Triangles by Sides You can classify a triangle by the measures of its sides.Equal numbers of hash marks indicate congruent sides.

• If all three sides of a triangle are congruent, then the triangle is an equilateral triangle.

• If at least two sides of a triangle are congruent, then the triangle is an isosceles triangle.

• If no two sides of a triangle are congruent, then the triangle is a scalene triangle.

Classify each triangle.

a. b. c.

Two sides are congruent. All three sides are The triangle has no pairThe triangle is an congruent. The triangle of congruent sides. It is isosceles triangle. is an equilateral triangle. a scalene triangle.

Classify each triangle as equilateral, isosceles, or scalene.

1. 2. 3.

4. 5. 6.

7. Find the measure of each side of equilateral �RST with RS � 2x � 2, ST � 3x,and TR � 5x � 4.

8. Find the measure of each side of isosceles �ABC with AB � BC if AB � 4y,BC � 3y � 2, and AC � 3y.

9. Find the measure of each side of �ABC with vertices A(�1, 5), B(6, 1), and C(2, �6).Classify the triangle.

D E

F

x

x x8x

32x

32x

B

CA

UW

S

12 17

19Q O

MG

K I18

18 182

1

3��G C

A

2312

15X V

TN

R PL J

H

Study Guide and Intervention (continued)

Classifying Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

ExampleExample

ExercisesExercises

Skills PracticeClassifying Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1


Less

on

4-1

Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.

1. 2. 3.

4. 5. 6.

Identify the indicated type of triangles.

7. right 8. isosceles

9. scalene 10. obtuse

ALGEBRA Find x and the measure of each side of the triangle.

11. �ABC is equilateral with AB� 3x � 2, BC � 2x � 4, and CA � x � 10.

12. �DEF is isosceles, �D is the vertex angle, DE � x � 7, DF � 3x � 1, and EF � 2x � 5.

Find the measures of the sides of �RST and classify each triangle by its sides.

13. R(0, 2), S(2, 5), T(4, 2)

14. R(1, 3), S(4, 7), T(5, 4)

E CD

A B


Use a protractor to classify each triangle as acute, equiangular, obtuse, or right.

1. 2. 3.

Identify the indicated type of triangles if A�B� � A�D� � B�D� � D�C�, B�E� � E�D�, A�B� ⊥ B�C�, and E�D� ⊥ D�C�.

4. right 5. obtuse

6. scalene 7. isosceles

ALGEBRA Find x and the measure of each side of the triangle.

8. �FGH is equilateral with FG � x � 5, GH � 3x � 9, and FH � 2x � 2.

9. �LMN is isosceles, �L is the vertex angle, LM � 3x � 2, LN � 2x � 1, and MN � 5x � 2.

Find the measures of the sides of �KPL and classify each triangle by its sides.

10. K(�3, 2) P(2, 1), L(�2, �3)

11. K(5, �3), P(3, 4), L(�1, 1)

12. K(�2, �6), P(�4, 0), L(3, �1)

13. DESIGN Diana entered the design at the right in a logo contest sponsored by a wildlife environmental group. Use a protractor.How many right angles are there?

A CD

E

B

Practice Classifying Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

Reading to Learn MathematicsClassifying Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1


Less

on

4-1

Pre-Activity Why are triangles important in construction?

Read the introduction to Lesson 4-1 at the top of page 178 in your textbook.

• Why are triangles used for braces in construction rather than other shapes?

• Why do you think that isosceles triangles are used more often thanscalene triangles in construction?

Reading the Lesson1. Supply the correct numbers to complete each sentence.

a. In an obtuse triangle, there are acute angle(s), right angle(s), and

obtuse angle(s).

b. In an acute triangle, there are acute angle(s), right angle(s), and

obtuse angle(s).

c. In a right triangle, there are acute angle(s), right angle(s), and

obtuse angle(s).

2. Determine whether each statement is always, sometimes, or never true.

a. A right triangle is scalene.

b. An obtuse triangle is isosceles.

c. An equilateral triangle is a right triangle.

d. An equilateral triangle is isosceles.

e. An acute triangle is isosceles.

f. A scalene triangle is obtuse.

3. Describe each triangle by as many of the following words as apply: acute, obtuse, right,scalene, isosceles, or equilateral.

a. b. c.

Helping You Remember4. A good way to remember a new mathematical term is to relate it to a nonmathematical

definition of the same word. How is the use of the word acute, when used to describeacute pain, related to the use of the word acute when used to describe an acute angle oran acute triangle?

5

34

135�80�

70�

30�


Reading MathematicsWhen you read geometry, you may need to draw a diagram to make the texteasier to understand.

Consider three points, A, B, and C on a coordinate grid.The y-coordinates of A and B are the same. The x-coordinate of B isgreater than the x-coordinate of A. Both coordinates of C are greaterthan the corresponding coordinates of B. Is triangle ABC acute, right,or obtuse?

To answer this question, first draw a sample triangle that fits the description.

Side AB must be a horizontal segment because the y-coordinates are the same. Point C must be located to the right and up from point B.

From the diagram you can see that triangle ABCmust be obtuse.

Answer each question. Draw a simple triangle on the grid above to help you.

1. Consider three points, R, S, and 2. Consider three noncollinear points,T on a coordinate grid. The J, K, and L on a coordinate grid. Thex-coordinates of R and S are the y-coordinates of J and K are thesame. The y-coordinate of T is same. The x-coordinates of K and Lbetween the y-coordinates of R are the same. Is triangle JKL acute,and S. The x-coordinate of T is less right, or obtuse?than the x-coordinate of R. Is angleR of triangle RST acute, right, or obtuse?

3. Consider three noncollinear points, 4. Consider three points, G, H, and ID, E, and F on a coordinate grid. on a coordinate grid. Points G and The x-coordinates of D and E are H are on the positive y-axis, andopposites. The y-coordinates of D and the y-coordinate of G is twice the E are the same. The x-coordinate of y-coordinate of H. Point I is on the F is 0. What kind of triangle must positive x-axis, and the x-coordinate�DEF be: scalene, isosceles, or of I is greater than the y-coordinateequilateral? of G. Is triangle GHI scalene,

isosceles, or equilateral?

BA

Q

x

y

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-14-1

ExampleExample

Study Guide and InterventionAngles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2

Angle Sum Theorem If the measures of two angles of a triangle are known,the measure of the third angle can always be found.

Angle Sum The sum of the measures of the angles of a triangle is 180.Theorem In the figure at the right, m�A � m�B � m�C � 180.

CA

B

Find m�T.

m�R � m�S � m�T � 180 Angle Sum

Theorem

25 � 35 � m�T � 180 Substitution

60 � m�T � 180 Add.

m�T � 120 Subtract 60

from each side.

35�

25�R T

S

Find the missing angle measures.

m�1 � m�A � m�B � 180 Angle Sum Theorem

m�1 � 58 � 90 � 180 Substitution

m�1 � 148 � 180 Add.

m�1 � 32 Subtract 148 from

each side.

m�2 � 32 Vertical angles are

congruent.

m�3 � m�2 � m�E � 180 Angle Sum Theorem

m�3 � 32 � 108 � 180 Substitution

m�3 � 140 � 180 Add.

m�3 � 40 Subtract 140 from

each side.

58�

90�

108�

12 3

E

DAC

B

Example 1Example 1 Example 2Example 2

ExercisesExercises

Find the measure of each numbered angle.

1. 2.

3. 4.

5. 6. 20�

152�

DG

A

130�60�

1 2

S

R

TW

Q

O

NM

P58�

66�

50�

321

V

W T

U

30�

60�

2

1

S

Q R30�

1

90�

62�

1 N

M

P


Exterior Angle Theorem At each vertex of a triangle, the angle formed by one sideand an extension of the other side is called an exterior angle of the triangle. For eachexterior angle of a triangle, the remote interior angles are the interior angles that are notadjacent to that exterior angle. In the diagram below, �B and �A are the remote interiorangles for exterior �DCB.

Exterior AngleThe measure of an exterior angle of a triangle is equal to

Theoremthe sum of the measures of the two remote interior angles.m�1 � m�A � m�B

AC

B

D1


Angles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

Find m�1.

m�1 � m�R � m�S Exterior Angle Theorem

� 60 � 80 Substitution

� 140 Add.

R T

S

60�

80�

1

Find x.

m�PQS � m�R � m�S Exterior Angle Theorem

78 � 55 � x Substitution

23 � x Subtract 55 from each side.

S R

Q

P

55�

78�

x�


ExercisesExercises

Find the measure of each numbered angle.

1. 2.

3. 4.

Find x.

5. 6. E

FGH

58�

x �

x �B

A

DC

95�

2x � 145�

U T

SRV

35� 36�

80�

13

2

POQ

N M60�

60�3 2

1

B C D

A

25�

35�

12Y Z W

X

65�

50�

1

Skills PracticeAngles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2


1. 2.

Find the measure of each angle.

3. m�1

4. m�2

5. m�3


6. m�1

7. m�2

8. m�3


9. m�1

10. m�2

11. m�3

12. m�4

13. m�5


14. m�1

15. m�2 63�

1

2D

A C

B

80�

60�

40�

105�

1 4 52

3

150�55�

70�

1 2

3

85� 55�

40�

1 2

3

146�

TIGERS80�

73�



1. 2.


3. m�1

4. m�2

5. m�3


6. m�1

7. m�4

8. m�3

9. m�2

10. m�5

11. m�6

Find the measure of each angle if �BAD and �BDC are right angles and m�ABC � 84.

12. m�1

13. m�2

14. CONSTRUCTION The diagram shows an example of the Pratt Truss used in bridgeconstruction. Use the diagram to find m�1.

145�1

64�1

2A

BC

D

118�36�

68�

70�

65�

82�

1

2

3 4

5

6

58�

39�

35�

12

3

40� 55�

72�

?

Practice Angles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

Reading to Learn MathematicsAngles of Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2


Less

on

4-2

Pre-Activity How are the angles of triangles used to make kites?


The frame of the simplest kind of kite divides the kite into four triangles.Describe these four triangles and how they are related to each other.

Reading the Lesson

1. Refer to the figure.

a. Name the three interior angles of the triangle. (Use threeletters to name each angle.)

b. Name three exterior angles of the triangle. (Use three lettersto name each angle.)

c. Name the remote interior angles of �EAB.

d. Find the measure of each angle without using a protractor.

i. �DBC ii. �ABC iii. �ACF iv. �EAB

2. Indicate whether each statement is true or false. If the statement is false, replace theunderlined word or number with a word or number that will make the statement true.

a. The acute angles of a right triangle are .

b. The sum of the measures of the angles of any triangle is .

c. A triangle can have at most one right angle or angle.

d. If two angles of one triangle are congruent to two angles of another triangle, then thethird angles of the triangles are .

e. The measure of an exterior angle of a triangle is equal to the of themeasures of the two remote interior angles.

f. If the measures of two angles of a triangle are 62 and 93, then the measure of thethird angle is .

g. An angle of a triangle forms a linear pair with an interior angle of thetriangle.

Helping You Remember

3. Many students remember mathematical ideas and facts more easily if they see themdemonstrated visually rather than having them stated in words. Describe a visual wayto demonstrate the Angle Sum Theorem.

exterior

35

difference

congruent

acute

100

supplementary

39�

23�

EA B D

CF


Finding Angle Measures in TrianglesYou can use algebra to solve problems involving triangles.

In triangle ABC, m�A, is twice m�B, and m�Cis 8 more than m�B. What is the measure of each angle?

Write and solve an equation. Let x � m�B.

m�A � m�B � m�C � 1802x � x � (x � 8) � 180

4x � 8 � 1804x � 172x � 43

So, m�A � 2(43) or 86, m�B � 43, and m�C � 43 � 8 or 51.

Solve each problem.

1. In triangle DEF, m�E is three times 2. In triangle RST, m�T is 5 more than m�D, and m�F is 9 less than m�E. m�R, and m�S is 10 less than m�T.What is the measure of each angle? What is the measure of each angle?

3. In triangle JKL, m�K is four times 4. In triangle XYZ, m�Z is 2 more than twicem�J, and m�L is five times m�J. m�X, and m�Y is 7 less than twice m�X.What is the measure of each angle? What is the measure of each angle?

5. In triangle GHI, m�H is 20 more than 6. In triangle MNO, m�M is equal to m�N,m�G, and m�G is 8 more than m�I. and m�O is 5 more than three times What is the measure of each angle? m�N. What is the measure of each angle?

7. In triangle STU, m�U is half m�T, 8. In triangle PQR, m�P is equal to and m�S is 30 more than m�T. What m�Q, and m�R is 24 less than m�P.is the measure of each angle? What is the measure of each angle?

9. Write your own problems about measures of triangles.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-24-2

ExampleExample

Study Guide and InterventionCongruent Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3


Less

on

4-3

Corresponding Parts of Congruent TrianglesTriangles that have the same size and same shape are congruent triangles. Two triangles are congruent if and only if all three pairs of corresponding angles are congruent and all three pairs of corresponding sides are congruent. In the figure, �ABC � �RST.

If �XYZ � �RST, name the pairs of congruent angles and congruent sides.�X � �R, �Y � �S, �Z � �TX�Y� � R�S�, X�Z� � R�T�, Y�Z� � S�T�

Identify the congruent triangles in each figure.

1. 2. 3.

Name the corresponding congruent angles and sides for the congruent triangles.

4. 5. 6. R

T

U S

B D

CA

F G L K

JE

K

J

L

MC

D

A

B

CA

B

LJ

K

Y

XZ

T

SR

AC

B

R

T

S

ExampleExample

ExercisesExercises


Identify Congruence Transformations If two triangles are congruent, you canslide, flip, or turn one of the triangles and they will still be congruent. These are calledcongruence transformations because they do not change the size or shape of the figure.It is common to use prime symbols to distinguish between an original �ABC and atransformed �A�B�C�.

Name the congruence transformation that produces �A�B�C� from �ABC.The congruence transformation is a slide.�A � �A�; �B � �B�; �C ��C�;A�B� � A��B��; A�C� � A��C��; B�C� � B��C��

Describe the congruence transformation between the two triangles as a slide, aflip, or a turn. Then name the congruent triangles.

1. 2.

3. 4.

5. 6.

x

y

OM

N

P

N�

P�

x

y

OA� B�

C�

A B

C

x

y

OB�B

A

Cx

y

O

Q� P�

P

Q

x

y

ON�

M�P�

N

MP

x

y

OR

S�T�

S

T

x

y

O

A�

B�B

C�A C


Congruent Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3

ExampleExample

ExercisesExercises

Skills PracticeCongruent Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3


Less

on

4-3


1. 2.

3. 4.

Name the congruent angles and sides for each pair of congruent triangles.

5. �ABC � �FGH

6. �PQR � �STU

Verify that each of the following transformations preserves congruence, and namethe congruence transformation.

7. �ABC � �A�B�C� 8. �DEF � �D�E�F �

x

y

OD�

E�

F�D

E

F

x

y

OA�

B�

C�

A

B

C

D

E

G

FRP

Q

S

WY

XC

AB

L

P

J

S

V

T



1. 2.

Name the congruent angles and sides for each pair of congruent triangles.

3. �GKP � �LMN

4. �ANC � �RBV

Verify that each of the following transformations preserves congruence, and namethe congruence transformation.

5. �PST � �P�S�T� 6. �LMN � �L�M�N�

QUILTING For Exercises 7 and 8, refer to the quilt design.

7. Indicate the triangles that appear to be congruent.

8. Name the congruent angles and congruent sides of a pair of congruent triangles.

B

A

I

E

FH G

C D

x

y

O

M�

N�L�

M

NL

x

y

O

S�

T�P�

S

TP

MN

L

P

QD

R

SCA

B

Practice Congruent Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3

Reading to Learn MathematicsCongruent Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3


Less

on

4-3

Pre-Activity Why are triangles used in bridges?


In the bridge shown in the photograph in your textbook, diagonal braceswere used to divide squares into two isosceles right triangles. Why do youthink these braces are used on the bridge?

Reading the Lesson1. If �RST � �UWV, complete each pair of congruent parts.

�R � � �W �T �

R�T� � � U�W� � W�V�

2. Identify the congruent triangles in each diagram.

a. b.

c. d.

3. Determine whether each statement says that congruence of triangles is reflexive,symmetric, or transitive.

a. If the first of two triangles is congruent to the second triangle, then the secondtriangle is congruent to the first.

b. If there are three triangles for which the first is congruent to the second and the secondis congruent to the third, then the first triangle is congruent to the third.

c. Every triangle is congruent to itself.

Helping You Remember4. A good way to remember something is to explain it to someone else. Your classmate Ben is

having trouble writing congruence statements for triangles because he thinks he has tomatch up three pairs of sides and three pairs of angles. How can you help him understandhow to write correct congruence statements more easily?

R T

US

V

N O P

QM

S

P R

Q

CA

B

D


Transformations in The Coordinate Plane

The following statement tells one way to map preimage points to image points in the coordinate plane.

(x, y) → (x � 6, y � 3)

This can be read, “The point with coordinates (x, y) is mapped to the point with coordinates (x � 6, y � 3).”With this transformation, for example, (3, 5) is mapped to (3 � 6, 5 � 3) or (9, 2). The figure shows how the triangle ABC is mapped to triangle XYZ.

1. Does the transformation above appear to be a congruence transformation? Explain youranswer.

Draw the transformation image for each figure. Then tell whether thetransformation is or is not a congruence transformation.

2. (x, y) → (x � 4, y) 3. (x, y) → (x � 8, y � 7)

4. (x, y) → (�x , �y) 5. (x, y) → ��12�x, y�

x

y

Ox

y

O

x

y

Ox

y

O

x

y

B

A

C X

Z

Y

O

(x, y ) → (x � 6, y � 3)

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-34-3

Study Guide and InterventionProving Congruence—SSS, SAS

NAME ______________________________________________ DATE ____________ PERIOD _____

4-44-4


Less

on

4-4

SSS Postulate You know that two triangles are congruent if corresponding sides arecongruent and corresponding angles are congruent. The Side-Side-Side (SSS) Postulate letsyou show that two triangles are congruent if you know only that the sides of one triangleare congruent to the sides of the second triangle.

SSS PostulateIf the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent.

Write a two-column proof.Given: A�B� � D�B� and C is the midpoint of A�D�.Prove: �ABC � �DBC

Statements Reasons

1. A�B� � D�B� 1. Given

2. C is the midpoint of A�D�. 2. Given

3. A�C� � D�C� 3. Definition of midpoint

4. B�C� � B�C� 4. Reflexive Property of �5. �ABC � �DBC 5. SSS Postulate

Write a two-column proof.

B

CDA

ExampleExample

ExercisesExercises

1.

Given: A�B� � X�Y�, A�C� � X�Z�, B�C� � Y�Z�Prove: �ABC � �XYZ

Statements Reasons

1.A�B� � X�Y� 1.Given

2.�ABC � �XYZ 2. SSS Post.

2.

Given: R�S� � U�T�, R�T� � U�S�Prove: �RST � �UTS

Statements Reasons

1.R�S� � U�T� 1. Given

2.S�T� � T�S� 2. Refl. Prop.3.�RST � �UTS 3. SSS Post.

T U

R S

B Y

CA XZ


SAS Postulate Another way to show that two triangles are congruent is to use the Side-Angle-Side (SAS) Postulate.

SAS PostulateIf two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

For each diagram, determine which pairs of triangles can beproved congruent by the SAS Postulate.

a. b. c.

In �ABC, the angle is not The right angles are The included angles, �1 “included” by the sides A�B� congruent and they are the and �2, are congruent and A�C�. So the triangles included angles for the because they are cannot be proved congruent congruent sides. alternate interior angles by the SAS Postulate. �DEF � �JGH by the for two parallel lines.

SAS Postulate. �PSR � �RQP by the SAS Postulate.

For each figure, determine which pairs of triangles can be proved congruent bythe SAS Postulate.

1. 2. 3.

4. 5. 6.

J H

GF

K

C

BA

D

V

T

W

M

P L

N

M

X

W Z

YQT

P

UN M

R

P Q

S

1

2 R

D H

F E

G

J

A

B C

X

Y Z


Proving Congruence—SSS, SAS

NAME ______________________________________________ DATE ____________ PERIOD _____

4-44-4

ExampleExample

ExercisesExercises

Skills PracticeProving Congruence—SSS, SAS

NAME ______________________________________________ DATE ____________ PERIOD _____

4-44-4


Less

on

4-4

Determine whether �ABC � �KLM given the coordinates of the vertices. Explain.

1. A(�3, 3), B(�1, 3), C(�3, 1), K(1, 4), L(3, 4), M(1, 6)

2. A(�4, �2), B(�4, 1), C(�1, �1), K(0, �2), L(0, 1), M(4, 1)

3. Write a flow proof.Given: P�R� � D�E�, P�T� � D�F�

�R � �E, �T � �FProve: �PRT � �DEF

Determine which postulate can be used to prove that the triangles are congruent.If it is not possible to prove that they are congruent, write not possible.

4. 5. 6.

PR � DEGiven

PT � DF Given

�R � �E Given

�P � �D Third AngleTheorem

�PRT � �DEF SAS

�T � �F Given

T

R

P

F

E

D


Determine whether �DEF � �PQR given the coordinates of the vertices. Explain.

1. D(�6, 1), E(1, 2), F(�1, �4), P(0, 5), Q(7, 6), R(5, 0)

2. D(�7, �3), E(�4, �1), F(�2, �5), P(2, �2), Q(5, �4), R(0, �5)

3. Write a flow proof.Given: R�S� � T�S�

V is the midpoint of R�T�.Prove: �RSV � �TSV

Determine which postulate can be used to prove that the triangles are congruent.If it is not possible to prove that they are congruent, write not possible.

4. 5. 6.

7. INDIRECT MEASUREMENT To measure the width of a sinkhole on his property, Harmon marked off congruent triangles as shown in thediagram. How does he know that the lengths A�B� and AB are equal?

A�B�

A B

C

RS � TSGiven

SV � SVReflexiveProperty

RV � VTDefinitionof midpoint

V is themidpoint of RT. Given

�RSV � �TSV SSS

S

R

V

T

Practice Proving Congruence—SSS, SAS

NAME ______________________________________________ DATE ____________ PERIOD _____

4-44-4

Reading to Learn MathematicsProving Congruence—SSS, SAS

NAME ______________________________________________ DATE ____________ PERIOD _____

4-44-4


Less

on

4-4

Pre-Activity How do land surveyors use congruent triangles?


Why do you think that land surveyors would use congruent right trianglesrather than other congruent triangles to establish property boundaries?

Reading the Lesson


a. Name the sides of �LMN for which �L is the included angle.

b. Name the sides of �LMN for which �N is the included angle.

c. Name the sides of �LMN for which �M is the included angle.

2. Determine whether you have enough information to prove that the two triangles in eachfigure are congruent. If so, write a congruence statement and name the congruencepostulate that you would use. If not, write not possible.

a. b.

c. E�H� and D�G� bisect each other. d.


3. Find three words that explain what it means to say that two triangles are congruent andthat can help you recall the meaning of the SSS Postulate.

GE

F

HD

R

T

SU

G

FD

E

C

A

DB

L

N

M


Congruent Parts of Regular Polygonal RegionsCongruent figures are figures that have exactly the same size and shape. There are manyways to divide regular polygonal regions into congruent parts. Three ways to divide anequilateral triangular region are shown. You can verify that the parts are congruent bytracing one part, then rotating, sliding, or reflecting that part on top of the other parts.

1. Divide each square into four congruent parts. Use three different ways.

2. Divide each pentagon into five congruent parts. Use three different ways.

3. Divide each hexagon into six congruent parts. Use three different ways.

4. What hints might you give another student who is trying to divide figures like those into congruent parts?

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-44-4

Study Guide and InterventionProving Congruence—ASA, AAS

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5


Less

on

4-5

ASA Postulate The Angle-Side-Angle (ASA) Postulate lets you show that two trianglesare congruent.

ASA PostulateIf two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

Find the missing congruent parts so that the triangles can beproved congruent by the ASA Postulate. Then write the triangle congruence.

a.

Two pairs of corresponding angles are congruent, �A � �D and �C � �F. If theincluded sides A�C� and D�F� are congruent, then �ABC � �DEF by the ASA Postulate.

b.

�R � �Y and S�R� � X�Y�. If �S � �X, then �RST� �YXW by the ASA Postulate.

What corresponding parts must be congruent in order to prove that the trianglesare congruent by the ASA Postulate? Write the triangle congruence statement.

1. 2. 3.

4. 5. 6.

A C

B

E

D

S

V

U

R T

D

A B

C

DC

EA

B

YW

X

ZE A

BD

C

R T W Y

S X

A C

B

D F

E

ExampleExample

ExercisesExercises


AAS Theorem Another way to show that two triangles are congruent is the Angle-Angle-Side (AAS) Theorem.

AAS TheoremIf two angles and a nonincluded side of one triangle are congruent to the corresponding twoangles and side of a second triangle, then the two triangles are congruent.

You now have five ways to show that two triangles are congruent.• definition of triangle congruence • ASA Postulate• SSS Postulate • AAS Theorem• SAS Postulate

In the diagram, �BCA � �DCA. Which sides are congruent? Which additional pair of corresponding parts needs to be congruent for the triangles to be congruent by the AAS Postulate?A�C� � A�C� by the Reflexive Property of congruence. The congruent angles cannot be �1 and �2, because A�C� would be the included side.If �B � �D, then �ABC � �ADC by the AAS Theorem.

In Exercises 1 and 2, draw and label �ABC and �DEF. Indicate which additionalpair of corresponding parts needs to be congruent for the triangles to becongruent by the AAS Theorem.

1. �A � �D; �B � �E 2. BC � EF; �A � �D

3. Write a flow proof.Given: �S � �U; T�R� bisects �STU.Prove: �SRT � �URT

Given

Given

RT � RT Refl. Prop. of �

Def.of � bisector

TR bisects �STU.

�SRT � �URT

�STR � �UTR

AAS�SRT � �URTCPCTC

�S � �U

S

R T

U

B

A

C

E

D

F

CA

B

FD

E

D

C12A

B


Proving Congruence—ASA, AAS

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5

ExampleExample

ExercisesExercises

Skills PracticeProving Congruence—ASA, AAS

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5


Less

on

4-5

Write a flow proof.

1. Given: �N � �LJ�K� � M�K�

Prove: �JKN � �MKL

2. Given: A�B� � C�B��A � �CD�B� bisects �ABC.

Prove: A�D� � C�D�

3. Write a paragraph proof.

Given: D�E� || F�G��E � �G

Prove: �DFG � �FDE

FG

D E

�A � �C

GivenAB � CB

Given CPCTCAD � CD

DB bisects �ABC. Given

�ABD � �CBDASA

�ABD � �CBDDef. of � bisector

A C

B

D

�N � �LGiven

JK � MK Given

�JKN � �MKL Vertical � are �.

�JKN � �MKLAAS

N

J

M

K L


1. Write a flow proof.Given: S is the midpoint of Q�T�.

Q�R� || T�U�Prove: �QSR � �TSU

2. Write a paragraph proof.

Given: �D � �FG�E� bisects �DEF.

Prove: D�G� � F�G�

ARCHITECTURE For Exercises 3 and 4, use the following information.An architect used the window design in the diagram when remodeling an art studio. A�B� and C�B� each measure 3 feet.

3. Suppose D is the midpoint of A�C�. Determine whether �ABD � �CBD.Justify your answer.

4. Suppose �A � �C. Determine whether �ABD � �CBD. Justify your answer.

D

B

A C

D

G

F

E

�Q � �T

Given

QR || TU Given

Def.of midpoint

Alt. Int. � are �.

QS � TS S is the midpoint of QT.

�QSR � �TSUASA

�QSR � �TSUVertical � are �.

UQ S

RT

Practice Proving Congruence—ASA, AAS

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5

Reading to Learn MathematicsProving Congruence—ASA, AAS

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5


Less

on

4-5

Pre-Activity How are congruent triangles used in construction?Read the introduction to Lesson 4-5 at the top of page 207 in your textbook.Which of the triangles in the photograph in your textbook appear to becongruent?

Reading the Lesson1. Explain in your own words the difference between how the ASA Postulate and the AAS

Theorem are used to prove that two triangles are congruent.

2. Which of the following conditions are sufficient to prove that two triangles are congruent?A. Two sides of one triangle are congruent to two sides of the other triangle.B. The three sides of one triangles are congruent to the three sides of the other triangle.C. The three angles of one triangle are congruent to the three angles of the other triangle.D. All six corresponding parts of two triangles are congruent.E. Two angles and the included side of one triangle are congruent to two sides and the

included angle of the other triangle.F. Two sides and a nonincluded angle of one triangle are congruent to two sides and a

nonincluded angle of the other triangle.G. Two angles and a nonincluded side of one triangle are congruent to two angles and

the corresponding nonincluded side of the other triangle.H. Two sides and the included angle of one triangle are congruent to two sides and the

included angle of the other triangle.I. Two angles and a nonincluded side of one triangle are congruent to two angles and a

nonincluded side of the other triangle.

3. Determine whether you have enough information to prove that the two triangles in eachfigure are congruent. If so, write a congruence statement and name the congruencepostulate or theorem that you would use. If not, write not possible.

a. b. T is the midpoint of R�U�.

Helping You Remember4. A good way to remember mathematical ideas is to summarize them in a general statement.

If you want to prove triangles congruent by using three pairs of corresponding parts,what is a good way to remember which combinations of parts will work?

R

S

T

U

V

A DCB

E


Congruent Triangles in the Coordinate PlaneIf you know the coordinates of the vertices of two triangles in the coordinateplane, you can often decide whether the two triangles are congruent. Theremay be more than one way to do this.

1. Consider � ABD and �CDB whose vertices have coordinates A(0, 0),B(2, 5), C(9, 5), and D(7, 0). Briefly describe how you can use what youknow about congruent triangles and the coordinate plane to show that � ABD � �CDB. You may wish to make a sketch to help get you started.

2. Consider �PQR and �KLM whose vertices are the following points.

P(1, 2) Q(3, 6) R(6, 5)K(�2, 1) L(�6, 3) M(�5, 6)

Briefly describe how you can show that �PQR � �KLM.

3. If you know the coordinates of all the vertices of two triangles, is it always possible to tell whether the triangles are congruent? Explain.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-54-5

Study Guide and InterventionIsosceles Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-64-6


Less

on

4-6

Properties of Isosceles Triangles An isosceles triangle has two congruent sides.The angle formed by these sides is called the vertex angle. The other two angles are calledbase angles. You can prove a theorem and its converse about isosceles triangles.

• If two sides of a triangle are congruent, then the angles opposite those sides are congruent. (Isosceles Triangle Theorem)

• If two angles of a triangle are congruent, then the sides opposite those angles are congruent. If A�B� � C�B�, then �A � �C.

If �A � �C, then A�B� � C�B�.

A

B

C

Find x.

BC � BA, so m�A � m�C. Isos. Triangle Theorem

5x � 10 � 4x � 5 Substitution

x � 10 � 5 Subtract 4x from each side.

x � 15 Add 10 to each side.

B

A

C (4x � 5)�

(5x � 10)�

Find x.

m�S � m�T, soSR � TR. Converse of Isos. � Thm.

3x � 13 � 2x Substitution

3x � 2x � 13 Add 13 to each side.

x � 13 Subtract 2x from each side.

R T

S

3x � 13

2x


ExercisesExercises

Find x.

1. 2. 3.

4. 5. 6.

7. Write a two-column proof.Given: �1 � �2Prove: A�B� � C�B�

Statements Reasons

B

A C D

E

1 32

R S

T3x�

x�D

BG

L3x�

30�D

T Q

PK

(6x � 6)� 2x�

W

Y Z3x�

S

V

T 3x � 6

2x � 6R

P

Q2x �

40�


Properties of Equilateral Triangles An equilateral triangle has three congruentsides. The Isosceles Triangle Theorem can be used to prove two properties of equilateraltriangles.

1. A triangle is equilateral if and only if it is equiangular.2. Each angle of an equilateral triangle measures 60°.

Prove that if a line is parallel to one side of an equilateral triangle, then it forms another equilateral triangle.Proof:Statements Reasons

1. �ABC is equilateral; P�Q� || B�C�. 1. Given2. m�A � m�B � m�C � 60 2. Each � of an equilateral � measures 60°.3. �1 � �B, �2 � �C 3. If || lines, then corres. �s are �.4. m�1 � 60, m�2 � 60 4. Substitution5. �APQ is equilateral. 5. If a � is equiangular, then it is equilateral.

Find x.

1. 2. 3.

4. 5. 6.

7. Write a two-column proof.Given: �ABC is equilateral; �1 � �2.Prove: �ADB � �CDB

Proof:

Statements Reasons

A

D

C

B12

R

O

HM 60�4x�

X

Z Y4x � 4

3x � 8 60�

P Q

LV R60�

4x 40

L

N

M

K

�KLM is equilateral.

3x�G

J H

6x � 5 5x

D

F E6x�

A

B

P Q

C

1 2


Isosceles Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-64-6

ExampleExample

ExercisesExercises

Skills PracticeIsosceles Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-64-6


Less

on

4-6

Refer to the figure.

1. If A�C� � A�D�, name two congruent angles.

2. If B�E� � B�C�, name two congruent angles.

3. If �EBA � �EAB, name two congruent segments.

4. If �CED � �CDE, name two congruent segments.

�ABF is isosceles, �CDF is equilateral, and m�AFD � 150.Find each measure.

5. m�CFD 6. m�AFB

7. m�ABF 8. m�A

In the figure, P�L� � R�L� and L�R� � B�R�.

9. If m�RLP � 100, find m�BRL.

10. If m�LPR � 34, find m�B.

11. Write a two-column proof.

Given: C�D� � C�G�D�E� � G�F�

Prove: C�E� � C�F�

DE

FG

C

R P

BL

D

C

F

B

35�

A E

D

C

B

A E


Refer to the figure.

1. If R�V� � R�T�, name two congruent angles.

2. If R�S� � S�V�, name two congruent angles.

3. If �SRT � �STR, name two congruent segments.

4. If �STV � �SVT, name two congruent segments.

Triangles GHM and HJM are isosceles, with G�H� � M�H�and H�J� � M�J�. Triangle KLM is equilateral, and m�HMK � 50.Find each measure.

5. m�KML 6. m�HMG 7. m�GHM

8. If m�HJM � 145, find m�MHJ.

9. If m�G � 67, find m�GHM.

10. Write a two-column proof.

Given: D�E� || B�C��1 � �2

Prove: A�B� � A�C�

11. SPORTS A pennant for the sports teams at Lincoln High School is in the shape of an isosceles triangle. If the measure of the vertex angle is 18, find the measure of each base angle. Lincoln Hawks

E

D B

C

A1

23

4

G

M

LK

J

H

U

R

TV

S

Practice Isosceles Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-64-6

Reading to Learn MathematicsIsosceles Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

4-64-6


Less

on

4-6

Pre-Activity How are triangles used in art?


• Why do you think that isosceles and equilateral triangles are used moreoften than scalene triangles in art?

• Why might isosceles right triangles be used in art?

Reading the Lesson1. Refer to the figure.

a. What kind of triangle is �QRS?

b. Name the legs of �QRS.

c. Name the base of �QRS.

d. Name the vertex angle of �QRS.

e. Name the base angles of �QRS.

2. Determine whether each statement is always, sometimes, or never true.

a. If a triangle has three congruent sides, then it has three congruent angles.

b. If a triangle is isosceles, then it is equilateral.

c. If a right triangle is isosceles, then it is equilateral.

d. The largest angle of an isosceles triangle is obtuse.

e. If a right triangle has a 45° angle, then it is isosceles.

f. If an isosceles triangle has three acute angles, then it is equilateral.

g. The vertex angle of an isosceles triangle is the largest angle of the triangle.

3. Give the measures of the three angles of each triangle.

a. an equilateral triangle

b. an isosceles right triangle

c. an isosceles triangle in which the measure of the vertex angle is 70

d. an isosceles triangle in which the measure of a base angle is 70

e. an isosceles triangle in which the measure of the vertex angle is twice the measure ofone of the base angles

Helping You Remember4. If a theorem and its converse are both true, you can often remember them most easily by

combining them into an “if-and-only-if” statement. Write such a statement for the IsoscelesTriangle Theorem and its converse.

R

Q

S


Triangle ChallengesSome problems include diagrams. If you are not sure how to solve theproblem, begin by using the given information. Find the measures of as manyangles as you can, writing each measure on the diagram. This may give youmore clues to the solution.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-64-6

1. Given: BE � BF, �BFG � �BEF ��BED, m�BFE � 82 andABFG and BCDE each haveopposite sides parallel andcongruent.

Find m�ABC.

3. Given: m�UZY � 90, m�ZWX � 45,�YZU � �VWX, UVXY is asquare (all sides congruent, allangles right angles).

Find m�WZY.

2. Given: AC � AD, and A�B��B�D�,m�DAC � 44 andC�E� bisects �ACD.

Find m�DEC.

4. Given: m�N � 120, J�N� � M�N�,�JNM � �KLM.

Find m�JKM.J

K

L

MN

A

D C

BE

U VW

XYZ

A

G DF E

CB

Study Guide and InterventionTriangles and Coordinate Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7


Less

on

4-7

Position and Label Triangles A coordinate proof uses points, distances, and slopes toprove geometric properties. The first step in writing a coordinate proof is to place a figure onthe coordinate plane and label the vertices. Use the following guidelines.

1. Use the origin as a vertex or center of the figure.2. Place at least one side of the polygon on an axis.3. Keep the figure in the first quadrant if possible.4. Use coordinates that make the computations as simple as possible.

Position an equilateral triangle on the coordinate plane so that its sides are a units long and one side is on the positive x-axis.Start with R(0, 0). If RT is a, then another vertex is T(a, 0).

For vertex S, the x-coordinate is �a2�. Use b for the y-coordinate,

so the vertex is S��a2�, b�.

Find the missing coordinates of each triangle.

1. 2. 3.

Position and label each triangle on the coordinate plane.

4. isosceles triangle 5. isosceles right �DEF 6. equilateral triangle �EQI�RST with base R�S� with legs e units long with vertex Q(0, a) and4a units long sides 2b units long

x

y

I(b, 0)E(–b, 0)

Q(0, a)

x

y

E(e, 0)

F(e, e)

D(0, 0)x

y T(2a, b)

R(0, 0) S(4a, 0)

x

y

G(2g, 0)

F(?, b)

E(?, ?)x

y

S(2a, 0)

T(?, ?)

R(0, 0)x

y

B(2p, 0)

C(?, q)

A(0, 0)

x

y

T(a, 0)R(0, 0)

S�a–2, b�

ExercisesExercises

ExampleExample


Write Coordinate Proofs Coordinate proofs can be used to prove theorems and toverify properties. Many coordinate proofs use the Distance Formula, Slope Formula, orMidpoint Theorem.

Prove that a segment from the vertex angle of an isosceles triangle to the midpoint of the base is perpendicular to the base.First, position and label an isosceles triangle on the coordinate plane. One way is to use T(a, 0), R(�a, 0), and S(0, c). Then U(0, 0) is the midpoint of R�T�.

Given: Isosceles �RST; U is the midpoint of base R�T�.Prove: S�U� ⊥ R�T�

Proof:U is the midpoint of R�T� so the coordinates of U are ��a

2� a�, �

0 �2

0�� (0, 0). Thus S�U� lies on

the y-axis, and �RST was placed so R�T� lies on the x-axis. The axes are perpendicular, so S�U� ⊥ R�T�.

Prove that the segments joining the midpoints of the sides of a right triangle forma right triangle.

C(2a, 0)

B(0, 2b)

P

Q

R

A(0, 0)

x

y

T(a, 0)U(0, 0)R(–a, 0)

S(0, c)


Triangles and Coordinate Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7

ExampleExample

ExercisesExercises

Skills PracticeTriangles and Coordinate Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7


Less

on

4-7


1. right �FGH with legs 2. isosceles �KLP with 3. isosceles �AND witha units and b units base K�P� 6b units long base A�D� 5a long


4. 5. 6.

7. 8. 9.

10. Write a coordinate proof to prove that in an isosceles right triangle, the segment fromthe vertex of the right angle to the midpoint of the hypotenuse is perpendicular to thehypotenuse.

Given: isosceles right �ABC with �ABC the right angle and M the midpoint of A�C�Prove: B�M� ⊥ A�C�

C(2a, 0)

A(0, 2a)

M

B(0, 0)

x

y

U(a, 0)

T(?, ?)

S(–a, 0)x

y

P(7b, 0)

R(?, ?)

N(0, 0)x

y

Q(?, ?)

R(2a, b)

P(0, 0)

x

y

N(3b, 0)

M(?, ?)

O(0, 0)x

y

Y(2b, 0)

Z(?, ?)

X(0, 0)x

y

B(2a, 0)

A(0, ?)

C(0, 0)

x

yN �5–

2a, b�

A(0, 0) D(5a, 0)x

y L(3b, c)

K(0, 0) P(6b, 0)x

y

F(0, a)

G(0, 0) H(b, 0)



1. equilateral �SWY with 2. isosceles �BLP with 3. isosceles right �DGJsides �

14�a long base B�L� 3b units long with hypotenuse D�J� and

legs 2a units long


4. 5. 6.

NEIGHBORHOODS For Exercises 7 and 8, use the following information.Karina lives 6 miles east and 4 miles north of her high school. After school she works parttime at the mall in a music store. The mall is 2 miles west and 3 miles north of the school.

7. Write a coordinate proof to prove that Karina’s high school, her home, and the mall areat the vertices of a right triangle.

Given: �SKMProve: �SKM is a right triangle.

8. Find the distance between the mall and Karina’s home.

x

y

S(0, 0)

K(6, 4)

M(–2, 3)

x

y

P(2b, 0)

M(0, ?)

N(?, 0)x

y

C(?, 0)

E(0, ?)

B(–3a, 0)x

y S(?, ?)

J(0, 0) R�1–3b, 0�

x

yD(0, 2a)

G(0, 0) J(2a, 0)x

yP �3–

2b, c�

B(0, 0) L(3b, 0)x

y Y �1–8a, b�

W �1–4a, 0�S(0, 0)

Practice Triangles and Coordinate Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7

Reading to Learn MathematicsTriangles and Coordinate Proof

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7


Less

on

4-7

Pre-Activity How can the coordinate plane be useful in proofs?


From the coordinates of A, B, and C in the drawing in your textbook, whatdo you know about �ABC?

Reading the Lesson

1. Find the missing coordinates of each triangle.

a. b.


a. Find the slope of S�R� and the slope of S�T�.

b. Find the product of the slopes of S�R� and S�T�. What does this tell you about S�R� and S�T�?

c. What does your answer from part b tell you about �RST ?

d. Find SR and ST. What does this tell you about S�R� and S�T�?

e. What does your answer from part d tell you about �RST?

f. Combine your answers from parts c and e to describe �RST as completely as possible.

g. Find m�SRT and m�STR.

h. Find m�OSR and m�OST.


3. Many students find it easier to remember mathematical formulas if they can put theminto words in a compact way. How can you use this approach to remember the slope andmidpoint formulas easily?

x

y

S(0, a)

R(–a, 0) T(a, 0)O(0, 0)

x

y

F(?, ?)E(?, a)

D(?, ?)x

y

T(a, ?)

R(?, b)

S(?, ?)


How Many Triangles?Each puzzle below contains many triangles. Count them carefully.Some triangles overlap other triangles.

How many triangles are there in each figure?

1. 2. 3.

4. 5. 6.

How many triangles can you form by joining points on each circle? List the vertices of each triangle.

7. 8.

8. 9. QR

P

U

S

TV

J K

O

L

MN

E F

I

GH

B

C

DA

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

4-74-7

Chapter 4 Test, Form 144


Ass

essm

ents

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

1. How would this triangle be classified by angles? A. acute B. equiangularC. obtuse D. right

2. What is the value of x if �ABC is equilateral?

A. �8 B. ��18�

C. �12� D. 2

Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right of each question.

3. What is m�2?A. 50 B. 70 C. 110 D. 120

4. What is m�4?A. 10 B. 60 C. 100 D. 120

5. What are the congruent triangles in the diagram? A. �ABC � �EBD B. �ABE � �CBDC. �AEB � �CBD D. �ABE � �CDB

6. If �CJW � �AGS, m�A � 50, m�J � 45,and m�S � 16x � 5, what is x? A. 17.5 B. 11.875C. 6 D. 5

7. Which postulate can be used to prove the triangles congruent? A. SSS B. SAS C. ASA D. AAS

8. What reason should be given for statement 5 in the proof?Given: D�B� is the perpendicular bisector of A�C�.Prove: �ADB � �CDB

Statements Reasons1. DB is the perpendicular bisector of A�C�. 1. Given2. A�B� � C�B� 2. Midpoint Theorem3. �ABD � �CBD 3. ⊥ line; all right � are �.4. D�B� � D�B� 4. Reflexive Property5. �ADB � �CDB 5.

A. SSS B. AAS C. ASA D. SAS

?

A C

D

B

C W

J

(16x � 5)�

45�

A S

G

50�

A

E

CB

D

A C

B

10x � 5

6x � 37.5x

1.

2.

3.

4.

5.

6.

7.

8.

NAME DATE PERIOD

SCORE

70�

2 13 460�

40�


Chapter 4 Test, Form 1 (continued)44

9.

10.

11.

12.

13.

Use the proof for Questions 9–10 and write the letter for the correct answer in the blank at theright of each question.

Given: L is the midpoint of J�M�; J�K� || N�M�.Prove: �JKL � �MNLStatements Reasons

1. L is the midpoint of J�M�. 1. Given2. J�L� � M�L� 2. Definition of midpoint3. J�K� || M�N� 3. Given4. �JKL � �MNL 4. Alt. int. � are �.5. �JLK � �MLN 5.6. �JKL � �MNL 6.

9. What is the reason for �JLK � �MLN?A. definition of midpointB. corresponding anglesC. vertical anglesD. alternate interior angles

10. What is the reason for �JKL � �MNL?A. AAS B. ASA C. SAS D. SSS

Use the figure for Questions 11–12 and write the letter for the correct answer in the blank at the right of each question.

11. If �LMN is isosceles and T is the midpoint of L�N�,which postulate can be used to prove �MLT � �MNT?A. AAA B. AAS C. SAS D. ABC

12. If �MLT � �MNT, what is used to prove �1 � �2?A. CPCTCB. definition of isosceles triangleC. definition of perpendicularD. definition of angle bisector

13. What are the missing coordinates of this triangle?A. (2a, 2c) B. (2a, 0)C. (0, 2a) D. (a, 2c)

Bonus What is the classification by sides of a triangle with coordinates A(5, 0), B(0, 5), and C(�5, 0)?

x

yM(a, c)

N(?, ?)L(0, 0)

1 2

M

TL N

(Question 10)(Question 9)

N

K

L MJ

B:

NAME DATE PERIOD

Chapter 4 Test, Form 2A44


Ass

essm

ents


1. What is the length of the sides of this equilateral triangle?A. 42 B. 30C. 15 D. 12

2. What is the classification of �ABC with vertices A(4, 1), B(2, �1), and C(�2, �1) by its sides?A. equilateral B. isosceles C. scalene D. right

Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right ofeach question.

3. What is m�1?A. 40 B. 50 C. 70 D. 90

4. What is m�3?A. 40 B. 70 C. 90 D. 110

5. If �DJL � �EGS, which segment in �EGS corresponds to D�L�?A. E�G� B. E�S� C. G�S� D. G�E�

6. Which triangles are congruent in the figure?A. �KLJ � �MNL B. �JLK � �NLMC. �JKL � �LMN D. �JKL � �MNL

Use the proof for Questions 7–8 and write the letter for the correct answer in the blank at the right of each question.

Given: R�J� || E�I�; R�I� bisects J�E�.Prove: �RJN � �IENStatements Reasons

1. R�J� || I�E� 1. Given2. �RJN � �IEN 2.3. R�I� bisects J�E�. 3. Given4. J�N� � E�N� 4. Definition of bisector5. �RNJ � �INE 5. Vert. � are �.6. �RJN � �IEN 6.

7. What is the reason for statement 2 in the proof?A. Isosceles Triangle Theorem B. same side interior anglesC. corresponding angles D. Alternate Interior Angle Theorem

8. What is the reason for statement 6?A. ASA B. AAS C. SAS D. SSS

(Question 8)

(Question 7)

E N

R

J

I

LK

NJ

M

70�

50�1

2

3

6x � 3

9x � 123x � 61.

2.

3.

4.

5.

6.

7.

8.

NAME DATE PERIOD

SCORE


Chapter 4 Test, Form 2A (continued)44

9.

10.

11.

12.

13.

9. If �ABC is isosceles and A�E� � F�C�, which theorem or postulate can be used to prove �AEB � �CFB? A. SSS B. SASC. ASA D. AAS


Given: D�A� || Y�N�; D�A� � Y�N�Prove: �NDY � �DNAStatements Reasons

1. D�A� || Y�N� 1. Given2. �ADN � �YND 2. Alt. int. � are �.3. D�A� � Y�N� 3. Given4. D�N� � D�N� 4. Reflexive Property5. �NDY � �DNA 5.6. �NDY � �DNA 6.

10. What is the reason for statement 5?A. ASA B. AASC. SAS D. SSS

11. What is the reason for statement 6?A. Alt. int. �s are �. B. CPCTCC. Corr. angles are �. D. Isosceles Triangle Theorem

12. What is the classification of a triangle with vertices A(3, 3), B(6, �2), C(0, �2)by its sides?A. isosceles B. scaleneC. equilateral D. right

13. What are the missing coordinates of the triangle?A. (�2b, 0) B. (0, 2b)C. (�c, 0) D. (0, �c)

Bonus Name the coordinates of points A and C in isosceles right �ABC if point Cis in the second quadrant.

x

y

B(0, a)

A(?, ?)

x

y(0, c)

(?, ?) (2b, 0)

(Question 11)(Question 10)

D A

NY

A C

B

E F

B:

NAME DATE PERIOD

Chapter 4 Test, Form 2B44


Ass

essm

ents


1. What is the length of the sides of this equilateral triangle? A. 2.5 B. 5C. 15 D. 20

2. What is the classification of �ABC with vertices A(0, 0), B(4, 3), and C(4, �3)by its sides?A. equilateral B. isosceles C. scalene D. right

Use the figure for Questions 3–4 and write the letter for the correct answer in the blank at the right ofeach question.

3. What is m�1?A. 120 B. 90 C. 60 D. 30

4. What is m�2?A. 120 B. 90 C. 60 D. 30

5. If �TGS � �KEL, which angle in �KEL corresponds to �T?A. �L B. �E C. �K D. �A

6. Which triangles are congruent in the figure?A. �HMN � �HGN B. �HMN � �NGHC. �NMH � �NGH D. �MNH � �HGN


Given: A�B� || C�D�; A�C� bisects B�D�.Prove: �ABE � �CDEStatements Reasons1. A�C� bisects B�D�. 1. Given2. B�E� � D�E� 2.3. A�B� || C�D� 3. Given4. �ABE � �CDE 4. Alt. int. � are �.5. 5.Vert. � are �.6. �ABE � �CDE 5. ASA

7. What is the reason for statement 2?A. Definition of bisector B. Midpoint TheoremC. Given D. Alternate Interior Angle Theorem

8. What is the statement for reason 5?A. �BEA � �DEC B. �ABE � �CDEC. �EAB � �ECD D. �BEC � �DEA

(Question 8)

(Question 7)

AB

CD

E

H G

NM

120�12

7x � 15

3x � 54x1.

2.

3.

4.

5.

6.

7.

8.

NAME DATE PERIOD

SCORE


Chapter 4 Test, Form 2B (continued)44

9.

10.

11.

12.

13.

9. If A�F� � D�E�, A�B� � F�C� and A�B� || F�C�, which theorem or postulate can be used to prove �ABE � �FCD?A. AAS B. ASAC. SAS D. SSS


Given: E�G� � I�A�; �EGA � �IAGProve: �GEN � �AINStatements Reasons

1. E�G� � I�A� 1. Given

2. �EGA � �IAG 2. Given

3. G�A� � G�A� 3. Reflexive Property

4. �EGA � �IAG 4.

5. �GEN � �AIN 5.

10. What is the reason for statement 4?A. SSS B. ASA C. SAS D. AAS

11. What is the reason for statement 5?A. Alt. int. � are �. B. Same Side Interior AnglesC. Corr. angles are �. D. CPCTC

12. What is the classification of a triangle with vertices A(�3, �1), B(�2, 2),C(3, 1) by its sides?A. scalene B. isoscelesC. equilateral D. right

13. What are the missing coordinates of the triangle?A. (a, 0) B. (b, 0)C. (c, 0) D. (0, c)

Bonus Find x in the triangle.(5x � 60)� (2x � 51)�

(30 � 10x)�(43 � 2x)�

x

y

(?, ?)

(a, 0)(�a, 0)

(Question 11)

(Question 10)

G A

N

E I

A DF E

CB

B:

NAME DATE PERIOD

Chapter 4 Test, Form 2C44


Ass

essm

ents

1. Use a protractor and ruler to classify the triangle by its angles and sides.

2. Find x, AB, BC, and AC if �ABC is equilateral.

3. Find the measure of the sides of the triangle if the vertices of�EFG are E(�3, 3), F(1, �1), and G(�3, �5). Then classify thetriangle by its sides.


4. m�1

5. m�2

6. m�3

7. Identify the congruent triangles and name their correspondingcongruent angles.

8. Verify that �ABC � �A�B�C�preserves congruence, assuming that corresponding angles arecongruent.

x

y

O

A�

B�

B

C�

AC

A

D F

GB

C

110� 2

1

3

A C

B

8x

7x � 310x � 6

1.

2.

3.

4.

5.

6.

7.

8.

NAME DATE PERIOD

SCORE


Chapter 4 Test, Form 2C (continued)44

9. ABCD is a quadrilateral with A�B� � C�D� and A�B� || C�D�. Name the postulate that could be used to prove �BAC � �DCA. Choose from SSS, SAS, ASA, and AAS.

10. �KLM is an isosceles triangle and �1 � �2. Name the theorem that could be used to determine �LKP � �LMN. Then name thepostulate that could be used to prove �LKP � �LMN. Choose from SSS, SAS, ASA, and AAS.

11. Use the figure to find m�1.

12. Find x.

13. Position and label isosceles �ABC with base A�B� b units longon the coordinate plane.

14. C�P� joins point C in isosceles right �ABC to the midpoint P, of A�B�.Name the coordinates of P. Thendetermine the relationship between A�B� and C�P�.

Bonus Without finding any other angles or sides congruent,which pair of triangles can be proved to be congruent bythe HL Theorem?

A

B

C D

E

F X

Y

Z M

N

O

x

yA(0, b)

B(b, 0)C(0, 0)

(10x � 20)�15x �

(18x � 12)�

40�

190�

1

P

1 2

N MK

L

A

D

B

C

NAME DATE PERIOD

9.

10.

11.

12.

13.

14.

B:

C(b–2, c)

B(b, 0)A

Chapter 4 Test, Form 2D44


Ass

essm

ents

1. Use a protractor and ruler to classify the triangle by its angles and sides.

2. Find x, AB, BC, AC if �ABC is isosceles.

3. Find the measure of the sides of the triangle if the vertices of�EFG are E(1, 4), F(5, 1), and G(2, �3). Then classify thetriangle by its sides.


4. m�1

5. m�2

6. m�3

7. Identify the congruent triangles and name their corresponding congruentangles.

8. Verify that �JKL � �J�K�L�preserves congruence, assuming that corresponding angles arecongruent.

9. In quadrilateral JKLM, J�K� � L�K�and M�K� bisects �LKJ. Name thepostulate that could be used to prove �MKL � �MKJ. Choose from SSS, SAS, ASA, and AAS.

M

K

J

L

x

y

O

K�J�

K

L�

J

L

DB

EC

FA

70�1 32

80�

A C

B

2x � 20

9x � 5

5x � 5

1.

2.

3.

4.

5.

6.

7.

8.

9.

NAME DATE PERIOD

SCORE


Chapter 4 Test, Form 2D (continued)44

10. �ABC is an isosceles triangle with B�D� ⊥ A�C�. Name the theorem that could be used to determine �A � �C. Then name the postulate that could be used to prove �BDA � �BDC. Choose from SSS, SAS, ASA, and AAS.

11. Use the figure to find m�1.

12. Find x.

13. Position and label equilateral �KLM with side lengths 3a units long on the coordinate plane.

14. M�N� joins the midpoint of A�B� and the midpoint of A�C� in �ABC. Findthe coordinates of M and N, and theslopes of M�N� and B�C�.

Bonus Without finding any other angles or sides congruent,which pair of triangles can be proved to be congruent by the LL Theorem?

A

B

C D

E

F X

Y

Z M

N

O

x

y

B(a, 0)

C(0, b)

M(?, ?)

N(?, ?)

A(0, 0)

(6x � 4)�

(18x � 8)�

80�1

B D

C

A

NAME DATE PERIOD

10.

11.

12.

13.

14.

B:

M(3a, 0)

L(1.5a, b)

K(0, 0)

Chapter 4 Test, Form 344


Ass

essm

ents

1. If �ABC is isosceles, �B is the vertex angle, AB � 20x � 2,BC � 12x � 30, and AC � 25x, find x and the measure of eachside of the triangle.

2. Given A(0, 4), B(5, 4), and C(�3, �2), find the measure of thesides of the triangle. Then classify the triangle by its sides and angles.

Use the figure to answer Questions 3–5.

3. Find x.

4. m�1, if m�1 � 4x � 10.

5. m�2

6. Verify that the following preserves congruence, assuming thatcorresponding angles are congruent. �ABC is reflected overthe x-axis as follows.A(�1, 1) → A�(�1, �1)B(4, 2) → B�(4, �2)C(1, 5) → C�(1, �5)Verify �ABC � �A�B�C�.

7. Determine whether �GHI � �JKL, given G(1, 2), H(5, 4),I(3, 6) and J(�4, �5), K(0, �3), L(�2, �1). Explain.

8. In the figure, A�C� � F�D�, A�B� || D�E�,and A�C� || F�D�. Name the postulate that could be used to prove �ABC � �DEC. Choose from SSS,SAS, ASA, and AAS.

D

E

A

B

F

C

(8x � 30)�(3x � 10)�2

1

1.

2.

3.

4.

5.

6.

7.

8.

NAME DATE PERIOD

SCORE


Chapter 4 Test, Form 3 (continued)44

For Questions 9 and 10, complete this two-column proof.

Given: �ABC is an isosceles triangle with base A�C�.D is the midpoint of A�C�.

Prove: B�D� bisects �ABC.

Statements Reasons1. �ABC is isosceles 1. Given

with base A�C�.

2. A�B� � C�B� 2. Def. of isosceles triangle.

3. �A � �C 3.

4. D is the midpoint of A�C�. 4. Given

5. A�D� � C�D� 5. Midpoint Theorem

6. �ABD � �CBD 6.

7. �1 � �2 7. CPCTC

8. B�D� bisects �ABC. 8. Def. of angle bisector

11. Find x.

12. Position and label isosceles �ABC with base A�B� (a � b) unitslong on a coordinate plane

Bonus In the figure, �ABC is isosceles, �ADC is equilateral,�AEC is isosceles, and the measures of �9, �1, and �3are all equal. Find the measures of the nine numberedangles.

987

1

65

23

4

B

C

DE

A

(15x � 15)�(21x � 3)�

(17x � 9)�

(Question 10)

(Question 9)

21

A C

B

D

NAME DATE PERIOD

9.

10.

11.

12.

B:

B(a � b, 0)

C(a � b, c)2

A(0, 0)

Chapter 4 Open-Ended Assessment44


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.

a. Classify the triangle by its angles and sides.

b. Show the steps needed to solve for x.

2. a. Describe how to determine whether a triangle with coordinates A(1, 4),B(1, �1), and C(�4, 4) is an equilateral triangle.

b. Is the triangle equilateral? Explain.

3. Explain how to find m�1 and m�2 in the figure.

4.

a. State the theorem or postulate that can be used to prove that thetriangles are congruent.

b. List their corresponding congruent angles and sides.

5.

Given: A�B� || D�E�, A�D� bisects B�E�.Prove: �ABC � �DEC by using the ASA postulate.

A

C

B

E D

J

DL

E

G

S

C

BD E

A 58�

40� 62�

1 2

(9x � 4)�

(20x � 10)�

NAME DATE PERIOD

SCORE


Chapter 4 Vocabulary Test/Review44

Choose from the terms above to complete each sentence.

1. A triangle that is equilateral is also called a(n) .

2. A(n) has at least one obtuse angle.

3. The sum of the is equivalent to the exterior angle of atriangle.

4. The angles of an isosceles triangle are congruent.

5. A triangle with different measures for each side is classified asa(n) .

6. A organizes a series of statements in logical orderwritten in boxes and uses arrows to indicate the order of thestatements.

7. A triangle that is translated, reflected or rotated and preservesits shape, is said to be a(n) .

8. The ASA postulate involves two corresponding angles andtheir corresponding .

9. A uses figures in the coordinate plane and algebra toprove geometric concepts.

10. The is formed by the congruent legs of an isoscelestriangle.

In your own words—

11. corollary

12. congruent triangles

13. acute triangle

?

?

?

?

?

?

?

?

?

? 1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

acute trianglebase anglescongruence

transformationscongruent triangles

coordinate proofcorollaryequiangular triangleequilateral triangleexterior angle

flow proofincluded angleincluded sideisosceles triangleobtuse triangle

remote interior anglesright trianglescalene trianglevertex angle

NAME DATE PERIOD

SCORE

Chapter 4 Quiz (Lessons 4–1 and 4–2)

44


Ass

essm

ents

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

1. Use a protractor to classify the triangle by its angles and sides.

2. STANDARDIZED TEST PRACTICE What is the best classification of this triangle by its angles and sides?A. acute isosceles B. right isoscelesC. obtuse isosceles D. obtuse equilateral

3. If �ABC is an isosceles triangle, �B is the vertex angle,AB � 6x � 3, BC � 8x � 1, and AC � 10x � 10, find x and themeasures of each side of the triangle.

4. If A(1, 5), B(3, �2), and C(�3, 0), find the measures of thesides of �ABC. Then classify the triangle by its sides.

Find the measure of each angle in the figure.

5. m�1 6. m�2

7. m�3 8. m�4

9. m�5 10. m�6

70�65�

1

2 34

56 107�

43�


44

1.

2.

3.

4.

1. Identify the congruent triangles in the figure.

2. STANDARDIZED TEST PRACTICE If �JGO � �RWI, whichangle corresponds to �I?A. �J B. �R C. �G D. �O

3. Verify that the following preserves congruence assuming that correspondingangles are congruent. �ABC � �A�B�C�

4. In quadrilateral EFGH, F�G� � H�E�, and F�G� || H�E�. Name the postulate that could be used to prove �EHF � �GFH. Choosefrom SSS, SAS, ASA, and AAS.

E H

GF

x

y

OC�

BB�

AC

A�

K

MLN

NAME DATE PERIOD

SCORE



44

1.

2.

3.

4.

For Questions 1 and 2, complete the two-column proof bysupplying the missing information for each correspondinglocation.

Given: �Z � �C; A�K� bisects �ZKC.Prove: �AKZ � �AKCStatements Reasons

1. �Z � �C; A�K� bisects �ZKC. 1. Given2. �ZKA � �CKA 2.3. A�K� � A�K� 3. Reflexive Property4. �AKZ � �AKC 4.

Refer to the figure for Questions 3 and 4.

3. Find m�1. 4. Find m�2.21

(Question 2)

(Question 1)

A

K

Z C

NAME DATE PERIOD

SCORE

Chapter 4 Quiz (Lesson 4–7)

44

1.

2.

3.

4.

5.

1. Find the missing coordinates.

Position and label each triangle on a coordinate plane.

2. Right �DJL with hypotenuse D�J�; LJ � �12�DL and D�L� is

a units long.

3. isosceles �EGS with base E�S� �12�b units long

For Questions 4 and 5, complete the coordinate proof bysupplying the missing information for each correspondinglocation.

Given: �ABC with A(�1, 1), B(5, 1), and C(2, 6).Prove: �ABC is isosceles.By the Distance Formula the lengths of the three sides are asfollows: . Since , �ABC is isosceles.(Question 5)(Question 4)

x

y

C(?, ?)

I(?, ?)

M(�b, 0)

S(1–2b, 0)

G(1–4b, c)

E(0, 0)

J(a–2, 0)

D(0, a)

L(0, 0)

NAME DATE PERIOD

SCORE

Chapter 4 Mid-Chapter Test (Lessons 4–1 through 4–3)

44


Ass

essm

ents

1. What is the best classification for this triangle?A. acute scaleneB. obtuse equilateralC. acute isoscelesD. obtuse isosceles


2. What is m�1?A. 50 B. 60C. 100 D. 105

3. What is m�2?A. 40 B. 50C. 60 D. 100

4. If �SJL � �DMT, which segment in �DMT corresponds to L�S� in �SJL?A. D�T� B. T�D�C. M�D� D. M�T�

50�

50�

60�

2

1

5.

6.

7.

NAME DATE PERIOD

SCORE

1.

2.

3.

4.

Part II

5. Find the measures of the sides of �ABC and classify it by itssides. A(1, 3), B(5, �2), and C(0, �4)

6. In �ABC and �A�B�C�, �A � �A�,�B � �B�, and �C � �C�. Find the lengths needed to prove �ABC � �A�B�C�.

7. What information would you need to know about P�O� and L�N� for �LMPto be congruent to �NMO by SSS? P O

N

L

M

x

y

O

C�

B

B�A

CA�

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


Chapter 4 Cumulative Review(Chapters 1–4)

44

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

1. Name the geometric figure that is modeled by the second handof a clock. (Lesson 1-1)

2. Find the precision for a measurement of 36 inches. (Lesson 1-2)

For Questions 3–5, use the number line.

3. Find BC. (Lesson 1-3)

4. Find the coordinate of the midpoint of A�D�. (Lesson 1-3)

5. If B is the midpoint of a segment having one endpoint at E,what is the coordinate of its other endpoint? (Lesson 1-3)

For Questions 6 and 7, determine whether each statementis always, sometimes, or never true. Explain your answer.(Lesson 2-5)

6. If D�E� � E�F�, then E is the midpoint of D�F�.

7. If points A and B lie in plane Q , then AB�� lies in Q .

8. Find the slope of a line parallel to x � 2. (Lesson 3-3)

9. Find the distance between y � �9 and y � �5. (Lesson 3-6)

For Questions 10–12, use the figure.

10. Name the segment that represents the distance from F to AD��. (Lesson 3-6)

11. Classify �ADC. (Lesson 4-1)

12. Find m�ACD. (Lesson 4-2)

13. Name the corresponding congruent angles and sides for �PQR � �HGB. (Lesson 4-3)

14. If �QRP � �SRT, and R is the midpoint of P�T�, which theorem or postulate can beused to prove �QRP � �SRT? Choose from SSS, SAS, ASA, and AAS. (Lesson 4-5)

15. Name the missing coordinates of �GEF. (Lesson 4-7)

x

y

F(2b, ?)D(0, 0) G(?, ?)

E(?, ?)

Q S

P TR

50�30�

85�

ED

A CB

F

�5 �4 �3 �2 �1�10 �9 �8 �7 �6 0 1 2 3 4 5 6

B ECA D

NAME DATE PERIOD

SCORE

Standardized Test Practice (Chapters 1–4)


1. If m�1 � 5x � 4, and m�2 � 52 � 9y, which values for x and ywould make �1 and �2 complementary? (Lesson 1-5)

A. x � 2, y � 12 B. x � 12, y � 2

C. x � 27, y � �13� D. x � �

13�, y � 27

2. Which is not a polygon? (Lesson 1-6)

E. F. G. H.

3. Complete the statement so that its conditional and its converseare true.If �1 � �2, then �1 and �2 . (Lesson 2-3)

A. are supplementary. B. are complementary.C. have the same measure. D. are alternate interior angles.

4. Complete this proof. (Lesson 2-7)

Given: U�V� � V�W�V�W� � W�X�

Prove: UV � WXProof:Statements Reasons

1. U�V� � V�W�; V�W� � W�X� 1. Given

2. UV � VW; VW � WX 2.3. UV � WX 3. Transitive Property

E. Definition of congruent segmentsF. Substitution PropertyG. Segment Addition PostulateH. Symmetric Property

5. Which equation has a slope of �13� and a y-intercept of �2? (Lesson 3-4)

A. y � �13�x � 2 B. y � �

13�x � 2

C. y � 2x � �13� D. y � �2x � �

13�

6. Classify �DEF with vertices D(2, 3), E(5, 7) and F(9, 4). (Lesson 4-1)

E. acute F. equiangular G. obtuse H. right

7. Which postulate or theorem can be used to prove �ABD � �CBD? (Lesson 4-4)

A. SAS B. SSSC. ASA D. AAS

A CB

D

?

U V

XW

?

NAME DATE PERIOD

SCORE 44

Part 1: Multiple Choice

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

1.

2.

3.

4.

5.

6.

7. 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


Standardized Test Practice (continued)

8. What is the y-coordinate of the midpoint ofA(12, 6) and B(�15, �6)? (Lesson 1-3)

9. If m�1 � 112, find m�10.(Lesson 3-2)

10. If J�K� || L�M�, then �4 must be supplementary to� . (Lesson 3-5)

11. Find PR if �PQR is isosceles, �Q is the vertexangle, PQ � 4x � 8, QR � x � 7, and PR � 6x � 12. (Lesson 4-1)

?

6

7

4

5H

L

M

J

K

�k

m

n

1 23

9 1011 12

45 6

7 8

NAME DATE PERIOD

44

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.

8. 9.

10. 11.

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

12. The perimeter of a regular pentagon is 14.5 feet. If each sidelength of the pentagon is doubled, what is the new perimeter?(Lesson 1-6)

13. Make a conjecture about the next number in the sequence 5, 7,11, 17, 25. (Lesson 2-1)

14. Find m�PQR. (Lesson 4-2)

15. If PQ � QS, QS � SR, and m�R � 20, find m�PSQ.(Lesson 4-6) P RS

Q

63� 125� 10�P S

Q

RT

12.

13.

14.

15.

0 6 8

6 1 8

Standardized Test PracticeStudent Record Sheet (Use with pages 232–233 of the Student Edition.)

44

© 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 12 and 14, also enter your answer by writing each number orsymbol in a box. Then fill in the corresponding oval for that number or symbol.

9 12 14

10

11

12 (grid in)

13

14 (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 15–16 on the back of this paper.


Stu

dy G

uid

e a

nd I

nte

rven

tion

Cla

ssif

yin

g T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1

©G

lenc

oe/M

cGra

w-H

ill18

3G

lenc

oe G

eom

etry

Lesson 4-1

Cla

ssif

y Tr

ian

gle

s b

y A

ng

les

On

e w

ay t

o cl

assi

fy a

tri

angl

e is

by

the

mea

sure

s of

its

an

gles

.

•If

one

of t

he a

ngle

s of

a t

riang

le is

an

obtu

se a

ngle

, th

en t

he t

riang

le is

an

ob

tuse

tri

ang

le.

•If

one

of t

he a

ngle

s of

a t

riang

le is

a r

ight

ang

le,

then

the

tria

ngle

is a

rig

ht

tria

ng

le.

•If

all t

hree

of t

he a

ngle

s of

a t

riang

le a

re a

cute

ang

les,

the

n th

e tr

iang

le is

an

acu

te t

rian

gle

.

•If

all t

hree

ang

les

of a

n ac

ute

tria

ngle

are

con

grue

nt,

then

the

tria

ngle

is a

n eq

uia

ng

ula

r tr

ian

gle

.

Cla

ssif

y ea

ch t

rian

gle.

a.

All

th

ree

angl

es a

re c

ongr

uen

t,so

all

th

ree

angl

es h

ave

mea

sure

60°

.T

he

tria

ngl

e is

an

equ

ian

gula

r tr

ian

gle.

b. T

he

tria

ngl

e h

as o

ne

angl

e th

at i

s ob

tuse

.It

is a

n o

btu

se t

rian

gle.

c.

Th

e tr

ian

gle

has

on

e ri

ght

angl

e.It

is

a ri

ght

tria

ngl

e.

Cla

ssif

y ea

ch t

rian

gle

as a

cute

,eq

uia

ngu

lar,

obtu

se,o

r ri

ght.

1.2.

3.

rig

ht

ob

tuse

equ

ian

gu

lar

4.5.

6.

acu

teri

gh

to

btu

se

60�

28�

92�

FDB

45�

45�

90�

XY

W

65�

65�

50�

UV

T

60�

60�

60�

Q

RS

120�

30�

30�

NO

P

67�

90�

23�

K LM

90�

60�

30�

G

HJ

25�

35�

120�

DF

E

60�A

BC

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill18

4G

lenc

oe G

eom

etry

Cla

ssif

y Tr

ian

gle

s b

y Si

des

You

can

cla

ssif

y a

tria

ngl

e by

th

e m

easu

res

of i

ts s

ides

.E

qual

nu

mbe

rs o

f h

ash

mar

ks i

ndi

cate

con

gru

ent

side

s.

•If

all t

hree

side

s of

a t

riang

le a

re c

ongr

uent

, th

en t

he t

riang

le is

an

equ

ilate

ral t

rian

gle

.

•If

at le

ast

two

side

s of

a t

riang

le a

re c

ongr

uent

, th

en t

he t

riang

le is

an

iso

scel

es t

rian

gle

.

•If

no t

wo

side

s of

a t

riang

le a

re c

ongr

uent

, th

en t

he t

riang

le is

a s

cale

ne

tria

ng

le.

Cla

ssif

y ea

ch t

rian

gle.

a.b

.c.

Tw

o si

des

are

con

gru

ent.

All

th

ree

side

s ar

e T

he

tria

ngl

e h

as n

o pa

irT

he

tria

ngl

e is

an

co

ngr

uen

t.T

he

tria

ngl

e of

con

gru

ent

side

s.It

is

isos

cele

s tr

ian

gle.

is a

n e

quil

ater

al t

rian

gle.

a sc

alen

e tr

ian

gle.

Cla

ssif

y ea

ch t

rian

gle

as e

qu

ila

tera

l,is

osce

les,

or s

cale

ne.

1.2.

3.

scal

ene

equ

ilate

ral

scal

ene

4.5.

6.

iso

scel

esis

osc

eles

equ

ilate

ral

7.F

ind

the

mea

sure

of

each

sid

e of

equ

ilat

eral

�R

ST

wit

h R

S�

2x�

2,S

T�

3x,

and

TR

�5x

�4.

2

8.F

ind

the

mea

sure

of

each

sid

e of

iso

scel

es �

AB

Cw

ith

AB

�B

Cif

AB

�4y

,B

C�

3y�

2,an

d A

C�

3y.

AB

�B

C�

8,A

C�

6

9.F

ind

the

mea

sure

of

each

sid

e of

�A

BC

wit

h v

erti

ces

A(�

1,5)

,B(6

,1),

and

C(2

,�6)

.C

lass

ify

the

tria

ngl

e.

AB

�B

C�

�65�

,AC

��

130

�;

�A

BC

is is

osc

eles

.

DE

Fx

xx

8x32

x

32x

B CA

UW

S

1217

19Q

O

MG

KI

18

1818

21

3�

�G

CA

2312

15X

V

TN

RP

LJ

H

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Cla

ssif

yin

g T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1

Exam

ple

Exam

ple

Exer

cises

Exer

cises

Answers (Lesson 4-1)


An

swer

s

Skil

ls P

ract

ice

Cla

ssif

yin

g T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1

©G

lenc

oe/M

cGra

w-H

ill18

5G

lenc

oe G

eom

etry

Lesson 4-1

Use

a p

rotr

acto

r to

cla

ssif

y ea

ch t

rian

gle

as a

cute

,eq

uia

ngu

lar,

obtu

se,o

r ri

ght.

1.2.

3.

equ

ian

gu

lar

ob

tuse

rig

ht

4.5.

6.

acu

teo

btu

seac

ute

Iden

tify

th

e in

dic

ated

typ

e of

tri

angl

es.

7.ri

ght

8.is

osce

les

�A

BE

,�B

CE

�B

CD

,�B

DE

9.sc

alen

e10

.obt

use

�A

BE

,�B

CE

�B

DE

ALG

EBR

AF

ind

xan

d t

he

mea

sure

of

each

sid

e of

th

e tr

ian

gle.

11.�

AB

Cis

equ

ilat

eral

wit

h A

B�

3x�

2,B

C�

2x�

4,an

d C

A�

x�

10.

x�

6,A

B�

16,B

C�

16,C

A�

16

12.�

DE

Fis

iso

scel

es,�

Dis

th

e ve

rtex

an

gle,

DE

�x

�7,

DF

�3x

�1,

and

EF

�2x

�5.

x�

4,D

E�

11,D

F�

11,E

F�

13

Fin

d t

he

mea

sure

s of

th

e si

des

of

�R

ST

and

cla

ssif

y ea

ch t

rian

gle

by

its

sid

es.

13.R

(0,2

),S

(2,5

),T

(4,2

)

RS

��

13�,S

T�

�13�

,RT

�4;

iso

scel

es

14.R

(1,3

),S

(4,7

),T

(5,4

)

RS

�5,

ST

��

10�,R

T�

�17�

;sc

alen

e

EC

D

AB

©G

lenc

oe/M

cGra

w-H

ill18

6G

lenc

oe G

eom

etry

Use

a p

rotr

acto

r to

cla

ssif

y ea

ch t

rian

gle

as a

cute

,eq

uia

ngu

lar,

obtu

se,o

r ri

ght.

1.2.

3.

ob

tuse

acu

teri

gh

t

Iden

tify

th

e in

dic

ated

typ

e of

tri

angl

es i

f A �

B��

A�D�

�B�

D��

D�C�

,B�E�

�E�

D�,A�

B�⊥

B�C�

,an

d E�

D�⊥

D�C�

.

4.ri

ght

5.ob

tuse

�A

BC

,�C

DE

�B

ED

,�B

DC

6.sc

alen

e7.

isos

cele

s

�A

BC

,�C

DE

�A

BD

,�B

ED

,�B

DC

AL

GE

BR

AF

ind

xan

d t

he

mea

sure

of

each

sid

e of

th

e tr

ian

gle.

8.�

FG

His

equ

ilat

eral

wit

h F

G�

x�

5,G

H�

3x�

9,an

d F

H�

2x�

2.

x�

7,F

G�

12,G

H�

12,F

H�

12

9.�

LM

Nis

iso

scel

es,�

Lis

the

ver

tex

angl

e,L

M�

3x�

2,L

N�

2x�

1,an

d M

N�

5x�

2.

x�

3,L

M�

7,L

N�

7,M

N�

13

Fin

d t

he

mea

sure

s of

th

e si

des

of

�K

PL

and

cla

ssif

y ea

ch t

rian

gle

by

its

sid

es.

10.K

(�3,

2) P

(2,1

),L

(�2,

�3)

KP

��

26�,P

L�

4�2�,

LK

��

26�;

iso

scel

es

11.K

(5,�

3),P

(3,4

),L

(�1,

1)

KP

��

53�,P

L�

5,L

K�

2�13�

;sc

alen

e

12.K

(�2,

�6)

,P(�

4,0)

,L(3

,�1)

KP

�2�

10�,P

L�

5�2�,

LK

�5�

2�;is

osc

eles

13.D

ESIG

ND

ian

a en

tere

d th

e de

sign

at

the

righ

t in

a l

ogo

con

test

sp

onso

red

by a

wil

dlif

e en

viro

nm

enta

l gr

oup.

Use

a p

rotr

acto

r.H

ow m

any

righ

t an

gles

are

th

ere?

5

AC

DE

B

Pra

ctic

e (

Ave

rag

e)

Cla

ssif

yin

g T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1



Readin

g t

o L

earn

Math

em

ati

csC

lass

ifyi

ng

Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1

©G

lenc

oe/M

cGra

w-H

ill18

7G

lenc

oe G

eom

etry

Lesson 4-1

Pre-

Act

ivit

yW

hy

are

tria

ngl

es i

mp

orta

nt

in c

onst

ruct

ion

?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-1

at

the

top

of p

age

178

in y

our

text

book

.

•W

hy a

re t

rian

gles

use

d fo

r br

aces

in c

onst

ruct

ion

rath

er t

han

othe

r sh

apes

?S

amp

le a

nsw

er:T

rian

gle

s lie

in a

pla

ne

and

are

rig

id s

hap

es.

•W

hy

do y

ou t

hin

k th

at i

sosc

eles

tri

angl

es a

re u

sed

mor

e of

ten

th

ansc

alen

e tr

ian

gles

in

con

stru

ctio

n?

Sam

ple

an

swer

:Is

osc

eles

tria

ng

les

are

sym

met

rica

l.

Rea

din

g t

he

Less

on

1.S

upp

ly t

he

corr

ect

nu

mbe

rs t

o co

mpl

ete

each

sen

ten

ce.

a.In

an

obt

use

tri

angl

e,th

ere

are

acu

te a

ngl

e(s)

,ri

ght

angl

e(s)

,an

d

obtu

se a

ngl

e(s)

.

b.

In a

n a

cute

tri

angl

e,th

ere

are

acu

te a

ngl

e(s)

,ri

ght

angl

e(s)

,an

d

obtu

se a

ngl

e(s)

.

c.In

a r

igh

t tr

ian

gle,

ther

e ar

e ac

ute

an

gle(

s),

righ

t an

gle(

s),a

nd

obtu

se a

ngl

e(s)

.

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.A

rig

ht

tria

ngl

e is

sca

len

e.so

met

imes

b.

An

obt

use

tri

angl

e is

iso

scel

es.

som

etim

esc.

An

equ

ilat

eral

tri

angl

e is

a r

igh

t tr

ian

gle.

nev

erd

.A

n e

quil

ater

al t

rian

gle

is i

sosc

eles

.al

way

se.

An

acu

te t

rian

gle

is i

sosc

eles

.so

met

imes

f.A

sca

len

e tr

ian

gle

is o

btu

se.

som

etim

es

3.D

escr

ibe

each

tri

angl

e by

as

man

y of

th

e fo

llow

ing

wor

ds a

s ap

ply:

acu

te,o

btu

se,r

igh

t,sc

alen

e,is

osce

les,

or e

quil

ater

al.

a.b

.c.

acu

te,s

cale

ne

ob

tuse

,iso

scel

esri

gh

t,sc

alen

e

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al t

erm

is

to r

elat

e it

to

a n

onm

ath

emat

ical

defi

nit

ion

of

the

sam

e w

ord.

How

is

the

use

of

the

wor

d ac

ute

,wh

en u

sed

to d

escr

ibe

acu

te p

ain

,rel

ated

to

the

use

of

the

wor

d ac

ute

wh

en u

sed

to d

escr

ibe

an a

cute

an

gle

oran

acu

te t

rian

gle?

Sam

ple

an

swer

:B

oth

are

rel

ated

to

th

e m

ean

ing

of

acu

teas

sh

arp

.An

acu

te p

ain

is a

sh

arp

pai

n,a

nd

an

acu

te a

ng

leca

n b

eth

ou

gh

t o

f as

an

an

gle

wit

h a

sh

arp

po

int.

In a

n a

cute

tri

ang

leal

l of

the

ang

les

are

acu

te.

5

34

135�

80�70�

30�

01

20

03

10

2

©G

lenc

oe/M

cGra

w-H

ill18

8G

lenc

oe G

eom

etry

Rea

din

g M

ath

emat

ics

Wh

en y

ou r

ead

geom

etry

,you

may

nee

d to

dra

w a

dia

gram

to

mak

e th

e te

xtea

sier

to

un

ders

tan

d.

Con

sid

er t

hre

e p

oin

ts,A

,B,a

nd

Con

a c

oord

inat

e gr

id.

Th

e y-

coor

din

ates

of

Aan

d B

are

the

sam

e.T

he

x-co

ord

inat

e of

Bis

grea

ter

than

th

e x-

coor

din

ate

of A

.Bot

h c

oord

inat

es o

f C

are

grea

ter

than

th

e co

rres

pon

din

g co

ord

inat

es o

f B

.Is

tria

ngl

e A

BC

acu

te,r

igh

t,or

ob

tuse

?

To

answ

er t

his

qu

esti

on,f

irst

dra

w a

sam

ple

tria

ngl

e

that

fit

s th

e de

scri

ptio

n.

Sid

e A

Bm

ust

be

a h

oriz

onta

l se

gmen

t be

cau

se t

he

y-co

ordi

nat

es a

re t

he

sam

e.P

oin

t C

mu

st b

e lo

cate

d to

th

e ri

ght

and

up

from

poi

nt

B.

Fro

m t

he

diag

ram

you

can

see

th

at t

rian

gle

AB

Cm

ust

be

obtu

se.

An

swer

eac

h q

ues

tion

.Dra

w a

sim

ple

tri

angl

e on

th

e gr

id a

bov

e to

hel

p y

ou.

1.C

onsi

der

thre

e po

ints

,R,S

,an

d 2.

Con

side

r th

ree

non

coll

inea

r po

ints

,T

on a

coo

rdin

ate

grid

.Th

e J

,K,a

nd

Lon

a c

oord

inat

e gr

id.T

he

x-co

ordi

nat

es o

f R

and

Sar

e th

ey-

coor

din

ates

of

Jan

d K

are

the

sam

e.T

he

y-co

ordi

nat

e of

Tis

sam

e.T

he

x-co

ordi

nat

es o

f K

and

Lbe

twee

n t

he

y-co

ordi

nat

es o

f R

are

the

sam

e.Is

tri

angl

e J

KL

acu

te,

and

S.T

he

x-co

ordi

nat

e of

Tis

les

sri

ght,

or o

btu

se?

rig

ht

than

th

e x-

coor

din

ate

of R

.Is

angl

eR

of t

rian

gle

RS

T a

cute

,rig

ht,

or

obtu

se?

acu

te

3.C

onsi

der

thre

e n

onco

llin

ear

poin

ts,

4.C

onsi

der

thre

e po

ints

,G,H

,an

d I

D,E

,an

d F

on a

coo

rdin

ate

grid

.on

a c

oord

inat

e gr

id.P

oin

ts G

and

Th

e x-

coor

din

ates

of

Dan

d E

are

Har

e on

th

e po

siti

ve y

-axi

s,an

dop

posi

tes.

Th

e y-

coor

din

ates

of

Dan

dth

e y-

coor

din

ate

of G

is t

wic

e th

e E

are

the

sam

e.T

he

x-co

ordi

nat

e of

y-co

ordi

nat

e of

H.P

oin

t I

is o

n t

he

Fis

0.W

hat

kin

d of

tri

angl

e m

ust

posi

tive

x-a

xis,

and

the

x-co

ordi

nat

e�

DE

F b

e:sc

alen

e,is

osce

les,

orof

Iis

gre

ater

th

an t

he

y-co

ordi

nat

eeq

uil

ater

al?

iso

scel

esof

G.I

s tr

ian

gle

GH

Isc

alen

e,is

osce

les,

or e

quil

ater

al?

scal

eneB

A

Q x

y

O

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-1

4-1

Exam

ple

Exam

ple



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

An

gle

s o

f Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2

©G

lenc

oe/M

cGra

w-H

ill18

9G

lenc

oe G

eom

etry

Lesson 4-2

An

gle

Su

m T

heo

rem

If t

he

mea

sure

s of

tw

o an

gles

of

a tr

ian

gle

are

know

n,

the

mea

sure

of

the

thir

d an

gle

can

alw

ays

be f

oun

d.

An

gle

Su

mT

he s

um o

f th

e m

easu

res

of t

he a

ngle

s of

a t

riang

le is

180

.T

heo

rem

In t

he f

igur

e at

the

rig

ht,

m�

A�

m�

B�

m�

C�

180.

CA

B

Fin

d m

�T

.

m�

R�

m�

S�

m�

T�

180

Ang

le S

um

The

orem

25 �

35 �

m�

T�

180

Sub

stitu

tion

60 �

m�

T�

180

Add

.

m�

T�

120

Sub

trac

t 60

from

eac

h si

de.

35�

25�

RT

S

Fin

d t

he

mis

sin

g an

gle

mea

sure

s.

m�

1 �

m�

A�

m�

B�

180

Ang

le S

um T

heor

em

m�

1 �

58 �

90�

180

Sub

stitu

tion

m�

1 �

148

�18

0A

dd.

m�

1�

32S

ubtr

act

148

from

each

sid

e.

m�

2�

32V

ertic

al a

ngle

s ar

e

cong

ruen

t.

m�

3 �

m�

2 �

m�

E�

180

Ang

le S

um T

heor

em

m�

3 �

32 �

108

�18

0S

ubst

itutio

n

m�

3 �

140

�18

0A

dd.

m�

3�

40S

ubtr

act

140

from

each

sid

e.

58�90

�

108�

12

3

E

DA

C

B

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

mea

sure

of

each

nu

mb

ered

an

gle.

1.m

�1

�28

2.m

�1

�12

0

3.4.

5.6.

m�

1 �

820

�

152�

DG

A

1

m�

1 �

30,

m�

2 �

6030

�60

�12

S

R

TW

m�

1 �

56,

m�

2 �

56,

m�

3 �

74Q

O

NM P

58�

66�

50�

32

1

m�

1 �

30,

m�

2 �

60V W

T

U

30�

60�

2

1

S

QR

30�

1

90�

62�

1NM

P

©G

lenc

oe/M

cGra

w-H

ill19

0G

lenc

oe G

eom

etry

Exte

rio

r A

ng

le T

heo

rem

At

each

ver

tex

of a

tri

angl

e,th

e an

gle

form

ed b

y on

e si

dean

d an

ext

ensi

on o

f th

e ot

her

sid

e is

cal

led

an e

xter

ior

angl

eof

th

e tr

ian

gle.

For

eac

hex

teri

or a

ngl

e of

a t

rian

gle,

the

rem

ote

inte

rior

an

gles

are

the

inte

rior

an

gles

th

at a

re n

otad

jace

nt

to t

hat

ext

erio

r an

gle.

In t

he

diag

ram

bel

ow,�

Ban

d �

Aar

e th

e re

mot

e in

teri

oran

gles

for

ext

erio

r �

DC

B.

Ext

erio

r A

ng

leT

he m

easu

re o

f an

ext

erio

r an

gle

of a

tria

ngle

is e

qual

to

Th

eore

mth

e su

m o

f th

e m

easu

res

of t

he t

wo

rem

ote

inte

rior

angl

es.

m�

1 �

m�

A�

m�

BA

C

B

D1

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

An

gle

s o

f Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2

Fin

d m

�1.

m�

1�

m�

R�

m�

SE

xter

ior

Ang

le T

heor

em

�60

�80

Sub

stitu

tion

�14

0A

dd.

RT

S

60�80

�

1

Fin

d x

.

m�

PQ

S�

m�

R�

m�

SE

xter

ior

Ang

le T

heor

em

78 �

55 �

xS

ubst

itutio

n

23 �

xS

ubtr

act

55 f

rom

eac

h si

de.

SR

Q

P

55�

78�

x�

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d t

he

mea

sure

of

each

nu

mb

ered

an

gle.

1.2.

m�

1 �

115

m�

1 �

60,m

�2

�12

0

3.4.

m�

1 �

60,m

�2

�60

,m�

3 �

120

m�

1 �

109,

m�

2 �

29,m

�3

�71

Fin

d x

.

5.25

6.29

E

FG

H58

�x�

x�B

A

DC

95�

2 x�

145�

UT

SR

V

35�

36�

80�

13

2

PO

Q

NM

60� 60

�3

2

1

BC

D

A

25�

35� 1

2Y

ZW

X 65�50

�

1



Skil

ls P

ract

ice

An

gle

s o

f Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2

©G

lenc

oe/M

cGra

w-H

ill19

1G

lenc

oe G

eom

etry

Lesson 4-2

Fin

d t

he

mis

sin

g an

gle

mea

sure

s.

1.27

2.17

,17

Fin

d t

he

mea

sure

of

each

an

gle.

3.m

�1

55

4.m

�2

55

5.m

�3

70

Fin

d t

he

mea

sure

of

each

an

gle.

6.m

�1

125

7.m

�2

55

8.m

�3

95

Fin

d t

he

mea

sure

of

each

an

gle.

9.m

�1

140

10.m

�2

40

11.m

�3

65

12.m

�4

75

13.m

�5

115

Fin

d t

he

mea

sure

of

each

an

gle.

14.m

�1

27

15.m

�2

2763

�

1

2D

AC

B

80�

60�

40�

105�

14

52

3

150�

55�

70�

12

3

85�

55�

40�

12

3

146�

TIG

ERS

80� 73�

©G

lenc

oe/M

cGra

w-H

ill19

2G

lenc

oe G

eom

etry

Fin

d t

he

mis

sin

g an

gle

mea

sure

s.

1.18

2.85

Fin

d t

he

mea

sure

of

each

an

gle.

3.m

�1

97

4.m

�2

83

5.m

�3

62

Fin

d t

he

mea

sure

of

each

an

gle.

6.m

�1

104

7.m

�4

45

8.m

�3

65

9.m

�2

79

10.m

�5

73

11.m

�6

147

Fin

d t

he

mea

sure

of

each

an

gle

if �

BA

Dan

d

�B

DC

are

righ

t an

gles

an

d m

�A

BC

�84

.

12.m

�1

26

13.m

�2

32

14.C

ON

STR

UC

TIO

NT

he

diag

ram

sh

ows

an

exam

ple

of t

he

Pra

tt T

russ

use

d in

bri

dge

con

stru

ctio

n.U

se t

he

diag

ram

to

fin

d m

�1.

55

145�

1

64�

1

2AB

C

D

118�

36�

68�

70� 65

� 82�

1

2

34

5

6

58�

39�

35�

12

3

40�

55�

72�

?

Pra

ctic

e (

Ave

rag

e)

An

gle

s o

f Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csA

ng

les

of T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2

©G

lenc

oe/M

cGra

w-H

ill19

3G

lenc

oe G

eom

etry

Lesson 4-2

Pre-

Act

ivit

yH

ow a

re t

he

angl

es o

f tr

ian

gles

use

d t

o m

ake

kit

es?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-2

at

the

top

of p

age

185

in y

our

text

book

.

Th

e fr

ame

of t

he

sim

ples

t ki

nd

of k

ite

divi

des

the

kite

in

to f

our

tria

ngl

es.

Des

crib

e th

ese

fou

r tr

ian

gles

an

d h

ow t

hey

are

rel

ated

to

each

oth

er.

Sam

ple

an

swer

:Th

ere

are

two

pai

rs o

f ri

gh

t tr

ian

gle

s th

at h

ave

the

sam

e si

ze a

nd

sh

ape.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.

a.N

ame

the

thre

e in

teri

or a

ngl

es o

f th

e tr

ian

gle.

(Use

th

ree

lett

ers

to n

ame

each

an

gle.

)�

BA

C,�

AB

C,�

BC

Ab

.N

ame

thre

e ex

teri

or a

ngl

es o

f th

e tr

ian

gle.

(Use

th

ree

lett

ers

to n

ame

each

an

gle.

)�

EA

B,�

DB

C,�

FC

Ac.

Nam

e th

e re

mot

e in

teri

or a

ngl

es o

f �

EA

B.

�A

BC

,�B

CA

d.

Fin

d th

e m

easu

re o

f ea

ch a

ngl

e w

ith

out

usi

ng

a pr

otra

ctor

.

i.�

DB

C62

ii.

�A

BC

118

iii.

�A

CF

157

iv.

�E

AB

141

2.In

dica

te w

het

her

eac

h s

tate

men

t is

tru

eor

fal

se.I

f th

e st

atem

ent

is f

alse

,rep

lace

th

eu

nde

rlin

ed w

ord

or n

um

ber

wit

h a

wor

d or

nu

mbe

r th

at w

ill

mak

e th

e st

atem

ent

tru

e.

a.T

he

acu

te a

ngl

es o

f a

righ

t tr

ian

gle

are

.fa

lse;

com

ple

men

tary

b.

Th

e su

m o

f th

e m

easu

res

of t

he

angl

es o

f an

y tr

ian

gle

is

.fa

lse;

180

c.A

tri

angl

e ca

n h

ave

at m

ost

one

righ

t an

gle

or

angl

e.fa

lse;

ob

tuse

d.

If t

wo

angl

es o

f on

e tr

ian

gle

are

con

gru

ent

to t

wo

angl

es o

f an

oth

er t

rian

gle,

then

th

eth

ird

angl

es o

f th

e tr

ian

gles

are

.

tru

ee.

Th

e m

easu

re o

f an

ext

erio

r an

gle

of a

tri

angl

e is

equ

al t

o th

e of

th

em

easu

res

of t

he

two

rem

ote

inte

rior

an

gles

.fa

lse;

sum

f.If

th

e m

easu

res

of t

wo

angl

es o

f a

tria

ngl

e ar

e 62

an

d 93

,th

en t

he

mea

sure

of

the

thir

d an

gle

is

.fa

lse;

25g.

An

an

gle

of a

tri

angl

e fo

rms

a li

nea

r pa

ir w

ith

an

in

teri

or a

ngl

e of

th

etr

ian

gle.

tru

e

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stu

den

ts r

emem

ber

mat

hem

atic

al i

deas

an

d fa

cts

mor

e ea

sily

if

they

see

th

emde

mon

stra

ted

visu

ally

rat

her

th

an h

avin

g th

em s

tate

d in

wor

ds.D

escr

ibe

a vi

sual

way

to d

emon

stra

te t

he

An

gle

Su

m T

heo

rem

.S

amp

le a

nsw

er:

Cu

t o

ff t

he

ang

les

of

a tr

ian

gle

an

d p

lace

th

em

sid

e-by

-sid

e o

n o

ne

sid

e o

f a

line

so t

hat

th

eir

vert

ices

mee

t at

a c

om

mo

np

oin

t.T

he

resu

lt w

ill s

ho

w t

hre

e an

gle

s w

ho

se m

easu

res

add

up

to

180

.

exte

rior

35

diff

eren

ce

con

gru

ent

acu

te

100

supp

lem

enta

ry

39�

23�

EA

BD

CF

©G

lenc

oe/M

cGra

w-H

ill19

4G

lenc

oe G

eom

etry

Fin

din

g A

ng

le M

easu

res

in T

rian

gle

sYo

u c

an u

se a

lgeb

ra t

o so

lve

prob

lem

s in

volv

ing

tria

ngl

es.

In t

rian

gle

AB

C,m

�A

,is

twic

e m

�B

,an

d m

�C

is 8

mor

e th

an m

�B

.Wh

at i

s th

e m

easu

re o

f ea

ch a

ngl

e?

Wri

te a

nd

solv

e an

equ

atio

n.L

et x

�m

�B

.

m�

A�

m�

B�

m�

C�

180

2x�

x�

(x�

8)�

180

4x�

8�

180

4x�

172

x�

43

So,

m�

A�

2(43

)or

86,m

�B

�43

,an

d m

�C

�43

�8

or51

.

Sol

ve e

ach

pro

ble

m.

1.In

tri

angl

e D

EF

,m�

E i

s th

ree

tim

es2.

In t

rian

gle

RS

T,m

�T

is 5

mor

e th

an

m�

D,a

nd

m�

Fis

9 l

ess

than

m�

E.

m�

R,a

nd

m�

Sis

10

less

th

an m

�T

.W

hat

is

the

mea

sure

of

each

an

gle?

Wh

at i

s th

e m

easu

re o

f ea

ch a

ngl

e?

m�

D�

27,m

�E

�81

,m�

F�

72m

�R

�60

,m�

S�

55,m

�T

�65

3.In

tri

angl

e J

KL

,m�

K i

s fo

ur

tim

es4.

In t

rian

gle

XY

Z,m

�Z

is 2

mor

e th

an t

wic

em

�J

,an

d m

�L

is f

ive

tim

es m

�J

.m

�X

,and

m�

Yis

7 l

ess

than

tw

ice

m�

X.

Wh

at i

s th

e m

easu

re o

f ea

ch a

ngl

e?W

hat

is

the

mea

sure

of

each

an

gle?

m�

J�

18,m

�K

�72

,m�

L�

90m

�X

�37

,m�

Y�

67,m

�Z

�76

5.In

tri

angl

e G

HI,

m�

H i

s 20

mor

e th

an6.

In t

rian

gle

MN

O,m

�M

is e

qual

to

m�

N,

m�

G,a

nd

m�

Gis

8 m

ore

than

m�

I.an

d m

�O

is 5

mor

e th

an t

hre

e ti

mes

W

hat

is

the

mea

sure

of

each

an

gle?

m�

N.W

hat

is

the

mea

sure

of

each

an

gle?

m�

G�

56,m

�H

�76

,m�

I�48

m�

M�

m�

N�

35,m

�O

�11

0

7.In

tri

angl

e S

TU

,m�

U i

s h

alf

m�

T,

8.In

tri

angl

e P

QR

,m�

Pis

equ

al t

o an

d m

�S

is 3

0 m

ore

than

m�

T.W

hat

m�

Q,a

nd

m�

Ris

24

less

th

an m

�P

.is

th

e m

easu

re o

f ea

ch a

ngl

e?W

hat

is

the

mea

sure

of

each

an

gle?

m�

S�

90,m

�T

�60

,m�

U�

30m

�P

�m

�Q

�68

,m�

R�

44

9.W

rite

you

r ow

n p

robl

ems

abou

t m

easu

res

of t

rian

gles

.S

ee s

tud

ents

’wo

rk.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-2

4-2

Exam

ple

Exam

ple



Stu

dy G

uid

e a

nd I

nte

rven

tion

Co

ng

ruen

t Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill19

5G

lenc

oe G

eom

etry

Lesson 4-3

Co

rres

po

nd

ing

Par

ts o

f C

on

gru

ent

Tria

ng

les

Tri

angl

es t

hat

hav

e th

e sa

me

size

an

d sa

me

shap

e ar

e co

ngr

uen

t tr

ian

gles

.Tw

o tr

ian

gles

are

con

gru

ent

if a

nd

only

if

all

thre

e pa

irs

of c

orre

spon

din

g an

gles

are

con

gru

ent

and

all

thre

e pa

irs

of c

orre

spon

din

g si

des

are

con

gru

ent.

In

the

figu

re,�

AB

C�

�R

ST

.

If �

XY

Z�

�R

ST

,nam

e th

e p

airs

of

con

gru

ent

angl

es a

nd

con

gru

ent

sid

es.

�X

��

R,�

Y�

�S

,�Z

��

TX �

Y��

R�S�

,X�Z�

�R�

T�,Y�

Z��

S�T�

Iden

tify

th

e co

ngr

uen

t tr

ian

gles

in

eac

h f

igu

re.

1.2.

3.

�A

BC

��

JKL

�A

BC

��

DC

B�

JKM

��

LM

K

Nam

e th

e co

rres

pon

din

g co

ngr

uen

t an

gles

an

d s

ides

for

th

e co

ngr

uen

t tr

ian

gles

.

4.5.

6.

�E

��

J;�

F�

�K

;�

A�

�D

;�

R�

�T

;�

G�

�L

;E�

F��

J�K�;

�A

BC

��

DC

B;

�R

SU

��

TS

U;

E�G�

�J�L�

;F�G�

�K�

L��

AC

B�

�D

BC

;�

RU

S�

�T

US

;A�

B��

D�C�

;A�

C��

D�B�

;R�

U��

T�U�;

R�S�

�T�S�

;B�

C��

C�B�

S�U�

�S�

U�

R TUS

BD

CA

FG

LK J

E

K J

L MC

D

A

B

CA

B

LJK

Y

XZ

T

SR

AC

B

R

TS

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill19

6G

lenc

oe G

eom

etry

Iden

tify

Co

ng

ruen

ce T

ran

sfo

rmat

ion

sIf

tw

o tr

ian

gles

are

con

gru

ent,

you

can

slid

e,fl

ip,o

r tu

rn o

ne

of t

he

tria

ngl

es a

nd

they

wil

l st

ill

be c

ongr

uen

t.T

hes

e ar

e ca

lled

con

gru

ence

tra

nsf

orm

atio

ns

beca

use

th

ey d

o n

ot c

han

ge t

he

size

or

shap

e of

th

e fi

gure

.It

is

com

mon

to

use

pri

me

sym

bols

to

dist

ingu

ish

bet

wee

n a

n o

rigi

nal

�A

BC

and

atr

ansf

orm

ed �

A�B

�C�.

Nam

e th

e co

ngr

uen

ce t

ran

sfor

mat

ion

th

at p

rod

uce

s �

A�B

�C�

from

�A

BC

.T

he

con

gru

ence

tra

nsf

orm

atio

n i

s a

slid

e.�

A�

�A

�;�

B�

�B

�;�

C�

�C

�;A �

B��

A��B�

��;A�

C��

A��C�

��;B�

C��

B��C�

��

Des

crib

e th

e co

ngr

uen

ce t

ran

sfor

mat

ion

bet

wee

n t

he

two

tria

ngl

es a

s a

slid

e,a

flip

,or

a tu

rn.T

hen

nam

e th

e co

ngr

uen

t tr

ian

gles

.

1.2.

flip

;�

RS

T�

�R

S�T

�sl

ide;

�M

NP

��

M�N

�P�

3.4.

turn

;�

OP

Q�

�O

P�Q

�fl

ip;

�A

BC

��

AB

�C

5.6.

slid

e;�

AB

C�

�A

�B�C

�tu

rn;

�M

NP

��

MN

�P�

x

y

OM

N

P

N�

P�

x

y

OA

�B

�

C�

ABC

x

y

OB

�B

A Cx

y

O

Q�

P�

P Q

x

y

ON

�

M�

P�

N MP

x

y

OR

S�

T�

S

T

x

y

O

A�

B�

B

C�

AC

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Co

ng

ruen

t Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Co

ng

ruen

t Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill19

7G

lenc

oe G

eom

etry

Lesson 4-3

Iden

tify

th

e co

ngr

uen

t tr

ian

gles

in

eac

h f

igu

re.

1.2.

�JP

L�

�T

VS

�A

BC

��

WX

Y

3.4.

�P

QR

��

PS

R�

DE

F�

�D

GF

Nam

e th

e co

ngr

uen

t an

gles

an

d s

ides

for

eac

h p

air

of c

ongr

uen

t tr

ian

gles

.

5.�

AB

C�

�F

GH

�A

��

F,�

B�

�G

,�C

��

H;

A�B�

�F�G�

,B�C�

�G�

H�,A�

C��

F�H�

6.�

PQ

R�

�S

TU

�P

��

S,�

Q�

�T

,�R

��

U;

P�Q�

�S�

T�,Q�

R��

T�U�,P�

R��

S�U�

Ver

ify

that

eac

h o

f th

e fo

llow

ing

tran

sfor

mat

ion

s p

rese

rves

con

gru

ence

,an

d n

ame

the

con

gru

ence

tra

nsf

orm

atio

n.

7.�

AB

C�

�A

�B�C

�8.

�D

EF

��

D�E

�F�

AB

�2�

2�,A

�B�

�2�

2�,D

E�

4,D

�E�

�4,

EF

�5,

BC

�2�

2�,B

�C�

�2�

2�,E

�F�

�5,

DF

� 3

,D�F

��

3,

AC

�4,

A�C

��

4,�

A�

�A

�,�

D�

�D

�,�

E�

�E

�,

�B

��

B�,

�C

��

C�;

slid

e�

F�

�F

�;fl

ip

x

y

OD

�

E�

F�

DE

F

x

y

OA

�

B�

C�

A

B

C

D

E G

FR

P

Q S

WY

XC

AB

L

P

J

S

V

T

©G

lenc

oe/M

cGra

w-H

ill19

8G

lenc

oe G

eom

etry

Iden

tify

th

e co

ngr

uen

t tr

ian

gles

in

eac

h f

igu

re.

1.2.

�A

BC

��

DR

S�

LM

N�

�Q

PN

Nam

e th

e co

ngr

uen

t an

gles

an

d s

ides

for

eac

h p

air

of c

ongr

uen

t tr

ian

gles

.

3.�

GK

P�

�L

MN

�G

��

L,�

K�

�M

,�P

��

N;

G�K�

�L�M�

,K�P�

�M�

N�,G�

P��

L�N�

4.�

AN

C�

�R

BV

�A

��

R,�

N�

�B

,�C

��

V;

A�N�

�R�

B�,N�

C��

B�V�

,A�C�

�R�

V�

Ver

ify

that

eac

h o

f th

e fo

llow

ing

tran

sfor

mat

ion

s p

rese

rves

con

gru

ence

,an

d n

ame

the

con

gru

ence

tra

nsf

orm

atio

n.

5.�

PS

T�

�P

�S�T

�6.

�L

MN

��

L�M

�N�

PS

��

13�,P

�S�

��

13�,

LM

�2�

2�,L

�M�

�2�

2�,S

T�

�5�,

S�T

��

�5�,

PT

��

10�,

MN

��

29�,M

�N�

��

29�,

P�T

��

�10�

,�P

��

P�,

LN

�7,

L�N

��

7,�

L�

�L

�,

�S

��

S�,

�T

��

T�;

flip

�M

��

M�,

�N

��

N�;

flip

QU

ILTI

NG

For

Exe

rcis

es 7

an

d 8

,ref

er t

o th

e q

uil

t d

esig

n.

7.In

dica

te t

he

tria

ngl

es t

hat

app

ear

to b

e co

ngr

uen

t.

�A

BI�

�E

BF

,�C

BD

��

HB

G

8.N

ame

the

con

gru

ent

angl

es a

nd

con

gru

ent

side

s of

a p

air

of

con

gru

ent

tria

ngl

es.

Sam

ple

an

swer

:�

A�

�E

,�A

BI�

�E

BF

,�I�

�F

;A�

B��

E�B�

,B�I��

B�F�,

A�I��

E�F�

B

A I

E FH

G

CD

x

y

O

M�

N�

L�

M

NL

x

y

O

S�

T�

P�

S

TP

MN

L

P

QD

R

SC

A

BPra

ctic

e (

Ave

rag

e)

Co

ng

ruen

t Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3


©G

lenc

oe/M

cGra

w-H

ill20

0G

lenc

oe G

eom

etry

Tran

sfo

rmat

ion

s in

Th

e C

oo

rdin

ate

Pla

ne

Th

e fo

llow

ing

stat

emen

t te

lls

one

way

to

map

pre

imag

e po

ints

to

imag

e po

ints

in

th

e co

ordi

nat

e pl

ane.

(x,y

) →

(x�

6,y

�3)

Th

is c

an b

e re

ad,“

Th

e po

int

wit

h c

oord

inat

es (

x,y)

is

map

ped

to t

he

poin

t w

ith

coo

rdin

ates

(x

�6,

y�

3).”

Wit

h t

his

tra

nsf

orm

atio

n,f

or e

xam

ple,

(3,5

) is

map

ped

to

(3 �

6,5

�3)

or

(9,2

).T

he

figu

re s

how

s h

ow t

he

tria

ngl

e A

BC

is m

appe

d to

tri

angl

e X

YZ

.

1.D

oes

the

tran

sfor

mat

ion

abo

ve a

ppea

r to

be

a co

ngr

uen

ce t

ran

sfor

mat

ion

? E

xpla

in y

our

answ

er.

Yes;

the

tran

sfo

rmat

ion

slid

es t

he

fig

ure

to

th

e lo

wer

rig

ht

wit

ho

ut

chan

gin

g it

s si

ze o

r sh

ape.

Dra

w t

he

tran

sfor

mat

ion

im

age

for

each

fig

ure

.Th

en t

ell

wh

eth

er t

he

tran

sfor

mat

ion

is

or i

s n

ot a

con

gru

ence

tra

nsf

orm

atio

n.

2.(x

,y)

→(x

�4,

y)ye

s3.

(x,y

) →

(x�

8,y

�7)

yes

4.(x

,y)

→(�

x,�

y)ye

s5.

(x,y

) →

��1 2� x

,y�

no

x

y

Ox

y

O

x

y

Ox

y

O

x

y

B

A

CX

ZY

O

(x, y

) → (x

� 6

, y �

3)

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3


Readin

g t

o L

earn

Math

em

ati

csC

on

gru

ent T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-3

4-3

©G

lenc

oe/M

cGra

w-H

ill19

9G

lenc

oe G

eom

etry

Lesson 4-3

Pre-

Act

ivit

yW

hy

are

tria

ngl

es u

sed

in

bri

dge

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-3

at

the

top

of p

age

192

in y

our

text

book

.

In t

he

brid

ge s

how

n i

n t

he

phot

ogra

ph i

n y

our

text

book

,dia

gon

al b

race

sw

ere

use

d to

div

ide

squ

ares

in

to t

wo

isos

cele

s ri

ght

tria

ngl

es.W

hy

do y

outh

ink

thes

e br

aces

are

use

d on

th

e br

idge

?S

amp

le a

nsw

er:T

he

dia

go

nal

bra

ces

mak

e th

e st

ruct

ure

str

on

ger

an

d p

reve

nt

itfr

om

bei

ng

def

orm

ed w

hen

it h

as t

o w

ith

stan

d a

hea

vy lo

ad.

Rea

din

g t

he

Less

on

1.If

�R

ST

��

UW

V,c

ompl

ete

each

pai

r of

con

gru

ent

part

s.

�R

��

�W

�T

�

R�T�

��

U�W�

�W�

V�

2.Id

enti

fy t

he

con

gru

ent

tria

ngl

es i

n e

ach

dia

gram

.

a.�

AB

C�

�A

DC

b.

�P

QS

��

RQ

S

c.d

.

�M

NO

��

QP

O�

RT

V�

�U

SV

3.D

eter

min

e w

het

her

eac

h s

tate

men

t sa

ys t

hat

con

gru

ence

of

tria

ngl

es i

s re

flex

ive,

sym

met

ric,

or t

ran

siti

ve.

a.If

th

e fi

rst

of t

wo

tria

ngl

es i

s co

ngr

uen

t to

th

e se

con

d tr

ian

gle,

then

th

e se

con

dtr

ian

gle

is c

ongr

uen

t to

th

e fi

rst.

sym

met

ric

b.

If t

here

are

thr

ee t

rian

gles

for

whi

ch t

he f

irst

is c

ongr

uent

to

the

seco

nd a

nd t

he s

econ

dis

con

gru

ent

to t

he

thir

d,th

en t

he

firs

t tr

ian

gle

is c

ongr

uen

t to

th

e th

ird.

tran

siti

vec.

Eve

ry t

rian

gle

is c

ongr

uen

t to

its

elf.

refl

exiv

e

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r so

met

hing

is t

o ex

plai

n it

to

som

eone

els

e.Yo

ur c

lass

mat

e B

en is

hav

ing

trou

ble

wri

tin

g co

ngr

uen

ce s

tate

men

ts f

or t

rian

gles

bec

ause

he

thin

ks h

e h

as t

om

atch

up

thre

e pa

irs

of s

ides

and

thr

ee p

airs

of

angl

es.H

ow c

an y

ou h

elp

him

und

erst

and

how

to

wri

te c

orre

ct c

ongr

uen

ce s

tate

men

ts m

ore

easi

ly?

Sam

ple

an

swer

:Wri

te t

he

thre

e ve

rtic

es o

f o

ne

tria

ng

le in

any

ord

er.T

hen

wri

te t

he

corr

esp

on

din

gve

rtic

es o

f th

e se

con

d t

rian

gle

in t

he

sam

e o

rder

.If

the

ang

les

are

wri

tten

in t

he

corr

ect

corr

esp

on

den

ce,t

he

sid

es w

ill a

uto

mat

ical

ly b

e in

th

eco

rrec

t co

rres

po

nd

ence

als

o.

RT

US

V

NO

P

QM

S

PR

Q

CA

B D

S�T�

R�S�

U�V�

�V

�S

�U



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Pro

vin

g C

on

gru

ence

—S

SS

,SA

S

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill20

1G

lenc

oe G

eom

etry

Lesson 4-4

SSS

Post

ula

teYo

u k

now

th

at t

wo

tria

ngl

es a

re c

ongr

uen

t if

cor

resp

ondi

ng

side

s ar

eco

ngr

uen

t an

d co

rres

pon

din

g an

gles

are

con

gru

ent.

Th

e S

ide-

Sid

e-S

ide

(SS

S)

Pos

tula

te l

ets

you

sh

ow t

hat

tw

o tr

ian

gles

are

con

gru

ent

if y

ou k

now

on

ly t

hat

th

e si

des

of o

ne

tria

ngl

ear

e co

ngr

uen

t to

th

e si

des

of t

he

seco

nd

tria

ngl

e.

SS

S P

ost

ula

teIf

the

side

s of

one

tria

ngle

are

con

grue

nt t

o th

e si

des

of a

sec

ond

tria

ngle

, th

en t

he t

riang

les

are

cong

ruen

t.

Wri

te a

tw

o-co

lum

n p

roof

.G

iven

:A �

B��

D�B�

and

Cis

th

e m

idpo

int

of A�

D�.

Pro

ve:�

AB

C�

�D

BC

Sta

tem

ents

Rea

son

s

1.A�

B��

D�B�

1.G

iven

2.C

is t

he

mid

poin

t of

A�D�

.2.

Giv

en

3.A�

C��

D�C�

3.D

efin

itio

n o

f m

idpo

int

4.B�

C��

B�C�

4.R

efle

xive

Pro

pert

y of

�5.

�A

BC

��

DB

C5.

SS

S P

ostu

late

Wri

te a

tw

o-co

lum

n p

roof

.

B CD

A

Exam

ple

Exam

ple

Exer

cises

Exer

cises

1.

Giv

en:A�

B��

X�Y�

,A�C�

�X�

Z�,B�

C��

Y�Z�

Pro

ve:�

AB

C�

�X

YZ

Sta

tem

ents

Rea

son

s

1.A�

B��

X�Y�

1.G

iven

A�C�

�X�

Z�B�

C��

Y�Z�

2.�

AB

C�

�X

YZ

2.S

SS

Po

st.

2.

Giv

en:R�

S��

U�T�

,R�T�

�U�

S�P

rove

:�R

ST

��

UT

S

Sta

tem

ents

Rea

son

s

1.R�

S��

U�T�

1.G

iven

R�T�

�U�

S�2.

S�T�

�T�S�

2.R

efl.

Pro

p.

3.�

RS

T�

�U

TS

3.S

SS

Po

st.

TU

RS

BY

CA

XZ

©G

lenc

oe/M

cGra

w-H

ill20

2G

lenc

oe G

eom

etry

SAS

Post

ula

teA

not

her

way

to

show

th

at t

wo

tria

ngl

es a

re c

ongr

uen

t is

to

use

th

e S

ide-

An

gle-

Sid

e (S

AS

) P

ostu

late

.

SA

S P

ost

ula

teIf

two

side

s an

d th

e in

clud

ed a

ngle

of

one

tria

ngle

are

con

grue

nt t

o tw

o si

des

and

the

incl

uded

ang

le o

f an

othe

r tr

iang

le,

then

the

tria

ngle

s ar

e co

ngru

ent.

For

eac

h d

iagr

am,d

eter

min

e w

hic

h p

airs

of

tria

ngl

es c

an b

ep

rove

d c

ongr

uen

t b

y th

e S

AS

Pos

tula

te.

a.b

.c.

In �

AB

C,t

he

angl

e is

not

T

he

righ

t an

gles

are

T

he

incl

ude

d an

gles

,�1

“in

clu

ded”

by t

he

side

s A�

B�co

ngr

uen

t an

d th

ey a

re t

he

and

�2,

are

con

gru

ent

an

d A �

C�.S

o th

e tr

ian

gles

in

clu

ded

angl

es f

or t

he

beca

use

th

ey a

re

can

not

be

prov

ed c

ongr

uen

t co

ngr

uen

t si

des.

alte

rnat

e in

teri

or a

ngl

es

by t

he

SA

S P

ostu

late

.�

DE

F�

�J

GH

by t

he

for

two

para

llel

lin

es.

SA

S P

ostu

late

.�

PS

R�

�R

QP

by t

he

SA

S P

ostu

late

.

For

eac

h f

igu

re,d

eter

min

e w

hic

h p

airs

of

tria

ngl

es c

an b

e p

rove

d c

ongr

uen

t b

yth

e S

AS

Pos

tula

te.

1.2.

3.

�T

RU

��

PM

Nby

th

e �

XQ

Yan

d �

WQ

Zar

e �

MP

L�

�N

PL

SA

S P

ost

ula

te.

no

t th

e in

clu

ded

an

gle

s b

ecau

se b

oth

are

fo

r th

e co

ng

ruen

t ri

gh

t an

gle

s.se

gm

ents

.Th

e tr

ian

gle

s �

MP

L�

�N

PL

by

are

no

t co

ng

ruen

t by

th

e S

AS

Po

stu

late

.th

e S

AS

Po

stu

late

.

4.5.

6.

Th

e tr

ian

gle

s ca

nn

ot

�D

��

Bb

ecau

se

Th

e co

ng

ruen

t b

e p

rove

d c

on

gru

ent

bo

th a

re r

igh

t an

gle

s.an

gle

s ar

e th

e by

th

e S

AS

Po

stu

late

.T

he

two

tri

ang

les

are

incl

ud

ed a

ng

les

for

con

gru

ent

by t

he

SA

S

the

con

gru

ent

sid

es.

Po

stu

late

.�

FJH

��

GH

Jby

the

SA

S P

ost

ula

te.

JH

GF

K

CBA D

V

T

W

M

PL

N M

X

WZY

QT

P

UN

MR

PQ

S

1

2R

DH

FE

G

J

A BC

X YZ

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Pro

vin

g C

on

gru

ence

—S

SS

,SA

S

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Pro

vin

g C

on

gru

ence

—S

SS

,SA

S

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill20

3G

lenc

oe G

eom

etry

Lesson 4-4

Det

erm

ine

wh

eth

er �

AB

C�

�K

LM

give

n t

he

coor

din

ates

of

the

vert

ices

.Exp

lain

.

1.A

(�3,

3),B

(�1,

3),C

(�3,

1),K

(1,4

),L

(3,4

),M

(1,6

)

AB

�2,

KL

�2,

BC

�2�

2�,L

M�

2�2�,

AC

�2,

KM

�2.

Th

e co

rres

po

nd

ing

sid

es h

ave

the

sam

e m

easu

re a

nd

are

co

ng

ruen

t,so

�A

BC

��

KL

Mby

SS

S.

2.A

(�4,

�2)

,B(�

4,1)

,C(�

1,�

1),K

(0,�

2),L

(0,1

),M

(4,1

)

AB

�3,

KL

�3,

BC

��

13�,L

M�

4,A

C�

�10�

,KM

�5.

Th

e co

rres

po

nd

ing

sid

es a

re n

ot

con

gru

ent,

so �

AB

Cis

no

t co

ng

ruen

t to

�K

LM

.

3.W

rite

a f

low

pro

of.

Giv

en:

P �R�

�D�

E�,P�

T��

D�F�

�R

��

E,�

T�

�F

Pro

ve:

�P

RT

��

DE

F

Pro

of:

Det

erm

ine

wh

ich

pos

tula

te c

an b

e u

sed

to

pro

ve t

hat

th

e tr

ian

gles

are

con

gru

ent.

If i

t is

not

pos

sib

le t

o p

rove

th

at t

hey

are

con

gru

ent,

wri

te n

ot p

ossi

ble.

4.5.

6.

SS

SS

AS

no

t p

oss

ible

PR �

DE

Give

n

PT �

DF

Give

n

�R

� �

E Gi

ven

�P

� �

D

Third

Ang

leTh

eore

m

�PR

T �

�D

EF

SAS

�T

� �

F Gi

ven

T R

P

F E

D

©G

lenc

oe/M

cGra

w-H

ill20

4G

lenc

oe G

eom

etry

Det

erm

ine

wh

eth

er �

DE

F�

�P

QR

give

n t

he

coor

din

ates

of

the

vert

ices

.Exp

lain

.

1.D

(�6,

1),E

(1,2

),F

(�1,

�4)

,P(0

,5),

Q(7

,6),

R(5

,0)

DE

�5�

2�,P

Q�

5�2�,

EF

�2�

10�,Q

R�

2�10�

,DF

�5�

2�,P

R�

5�2�.

�D

EF

��

PQ

Rby

SS

S s

ince

co

rres

po

nd

ing

sid

es h

ave

the

sam

em

easu

re a

nd

are

co

ng

ruen

t.

2.D

(�7,

�3)

,E(�

4,�

1),F

(�2,

�5)

,P(2

,�2)

,Q(5

,�4)

,R(0

,�5)

DE

��

13�,P

Q�

�13

,�

EF

�2�

5�,Q

R�

�26�

,DF

��

29�,P

R�

�13�

.C

orr

esp

on

din

g s

ides

are

no

t co

ng

ruen

t,so

�D

EF

is n

ot

con

gru

ent

to �

PQ

R.

3.W

rite

a f

low

pro

of.

Giv

en:

R �S�

�T�

S�V

is t

he

mid

poin

t of

R�T�

.P

rove

:�

RS

V�

�T

SV

Pro

of:

Det

erm

ine

wh

ich

pos

tula

te c

an b

e u

sed

to

pro

ve t

hat

th

e tr

ian

gles

are

con

gru

ent.

If i

t is

not

pos

sib

le t

o p

rove

th

at t

hey

are

con

gru

ent,

wri

te n

ot p

ossi

ble.

4.5.

6.

no

t p

oss

ible

SA

S o

r S

SS

SS

S

7.IN

DIR

ECT

MEA

SUR

EMEN

TT

o m

easu

re t

he

wid

th o

f a

sin

khol

e on

h

is p

rope

rty,

Har

mon

mar

ked

off

con

gru

ent

tria

ngl

es a

s sh

own

in

th

edi

agra

m.H

ow d

oes

he

know

th

at t

he

len

gth

s A�

B�

and

AB

are

equ

al?

Sin

ce �

AC

Ban

d �

A�C

B�

are

vert

ical

an

gle

s,th

ey a

re

con

gru

ent.

In t

he

fig

ure

,A�C�

�A�

��C�an

d B�

C��

B��C�

.So

�

AB

C�

�A

�B�C

by S

AS

.By

CP

CT

C,t

he

len

gth

s A

�B�

and

AB

are

equ

al.A

�B

�

AB

C

RS

� T

SGi

ven

SV �

SV

Refle

xive

Prop

erty

RV

� V

TDe

finiti

onof

mid

poin

t

V is

th

em

idp

oin

t o

f R

T.

Give

n

�R

SV �

�TS

V

SSS

S

R V T

Pra

ctic

e (

Ave

rag

e)

Pro

vin

g C

on

gru

ence

—S

SS

,SA

S

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csP

rovi

ng

Co

ng

ruen

ce—

SS

S,S

AS

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4

©G

lenc

oe/M

cGra

w-H

ill20

5G

lenc

oe G

eom

etry

Lesson 4-4

Pre-

Act

ivit

yH

ow d

o la

nd

su

rvey

ors

use

con

gru

ent

tria

ngl

es?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-4

at

the

top

of p

age

200

in y

our

text

book

.

Wh

y do

you

th

ink

that

lan

d su

rvey

ors

wou

ld u

se c

ongr

uen

t ri

ght

tria

ngl

esra

ther

th

an o

ther

con

gru

ent

tria

ngl

es t

o es

tabl

ish

pro

pert

y bo

un

dari

es?

Sam

ple

an

swer

:L

and

is u

sual

ly d

ivid

ed in

to r

ecta

ng

ula

r lo

ts,

so t

hei

r b

ou

nd

arie

s m

eet

at r

igh

t an

gle

s.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.

a.N

ame

the

side

s of

�L

MN

for

wh

ich

�L

is t

he

incl

ude

d an

gle.

L�M�,L�

N�b

.N

ame

the

side

s of

�L

MN

for

wh

ich

�N

is t

he

incl

ude

d an

gle.

N�L�,

N�M�

c.N

ame

the

side

s of

�L

MN

for

wh

ich

�M

is t

he

incl

ude

d an

gle.

M�L�,

M�N�

2.D

eter

min

e w

het

her

you

hav

e en

ough

in

form

atio

n t

o pr

ove

that

th

e tw

o tr

ian

gles

in

eac

hfi

gure

are

con

gru

ent.

If s

o,w

rite

a c

ongr

uen

ce s

tate

men

t an

d n

ame

the

con

gru

ence

post

ula

te t

hat

you

wou

ld u

se.I

f n

ot,w

rite

not

pos

sibl

e.

a.b

.

�A

BD

��

CB

D;

SA

Sn

ot

po

ssib

le

c.E�

H�an

d D�

G�bi

sect

eac

h o

ther

.d

.

�D

EF

��

GH

F;

SA

S�

RS

U�

�T

SU

;S

SS

Hel

pin

g Y

ou

Rem

emb

er

3.F

ind

thre

e w

ords

th

at e

xpla

in w

hat

it

mea

ns

to s

ay t

hat

tw

o tr

ian

gles

are

con

gru

ent

and

that

can

hel

p yo

u r

ecal

l th

e m

ean

ing

of t

he

SS

S P

ostu

late

.S

amp

le a

nsw

er:

Co

ng

ruen

t tr

ian

gle

s ar

e tr

ian

gle

s th

at a

re t

he

sam

e si

zean

d s

hap

e,an

d t

he

SS

S P

ost

ula

te e

nsu

res

that

tw

o t

rian

gle

s w

ith

th

ree

corr

esp

on

din

g s

ides

co

ng

ruen

t w

ill b

e th

e sa

me

size

an

d s

hap

e.

GE

F

HD

R T

SU

G

FD

E

CA

DB

L

N

M

©G

lenc

oe/M

cGra

w-H

ill20

6G

lenc

oe G

eom

etry

Co

ng

ruen

t P

arts

of

Reg

ula

r P

oly

go

nal

Reg

ion

sC

ongr

uen

t fi

gure

s ar

e fi

gure

s th

at h

ave

exac

tly

the

sam

e si

ze a

nd

shap

e.T

her

e ar

e m

any

way

s to

div

ide

regu

lar

poly

gon

al r

egio

ns

into

con

gru

ent

part

s.T

hre

e w

ays

to d

ivid

e an

equ

ilat

eral

tri

angu

lar

regi

on a

re s

how

n.Y

ou c

an v

erif

y th

at t

he

part

s ar

e co

ngr

uen

t by

trac

ing

one

part

,th

en r

otat

ing,

slid

ing,

or r

efle

ctin

g th

at p

art

on t

op o

f th

e ot

her

par

ts.

1.D

ivid

e ea

ch s

quar

e in

to f

our

con

gru

ent

part

s.U

se t

hre

e di

ffer

ent

way

s.S

amp

le a

nsw

ers

are

sho

wn

.

2.D

ivid

e ea

ch p

enta

gon

in

to f

ive

con

gru

ent

part

s.U

se t

hre

e di

ffer

ent

way

s.S

amp

le a

nsw

ers

are

sho

wn

.

3.D

ivid

e ea

ch h

exag

on i

nto

six

con

gru

ent

part

s.U

se t

hre

e di

ffer

ent

way

s.S

amp

le a

nsw

ers

are

sho

wn

.

4.W

hat

hin

ts m

igh

t yo

u g

ive

anot

her

stu

den

t w

ho

is t

ryin

g to

div

ide

figu

res

like

th

ose

into

con

gru

ent

part

s?S

ee s

tud

ents

’wo

rk.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-4

4-4



Stu

dy G

uid

e a

nd I

nte

rven

tion

Pro

vin

g C

on

gru

ence

—A

SA

,AA

S

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5

©G

lenc

oe/M

cGra

w-H

ill20

7G

lenc

oe G

eom

etry

Lesson 4-5

ASA

Po

stu

late

Th

e A

ngl

e-S

ide-

An

gle

(AS

A)

Pos

tula

te l

ets

you

sh

ow t

hat

tw

o tr

ian

gles

are

con

gru

ent.

AS

A P

ost

ula

teIf

two

angl

es a

nd t

he in

clud

ed s

ide

of o

ne t

riang

le a

re c

ongr

uent

to

two

angl

es

and

the

incl

uded

sid

e of

ano

ther

tria

ngle

, th

en t

he t

riang

les

are

cong

ruen

t.

Fin

d t

he

mis

sin

g co

ngr

uen

t p

arts

so

that

th

e tr

ian

gles

can

be

pro

ved

con

gru

ent

by

the

AS

A P

ostu

late

.Th

en w

rite

th

e tr

ian

gle

con

gru

ence

.

a.

Tw

o pa

irs

of c

orre

spon

din

g an

gles

are

con

gru

ent,

�A

��

Dan

d �

C�

�F

.If

the

incl

ude

d si

des

A�C�

and

D�F�

are

con

gru

ent,

then

�A

BC

��

DE

Fby

th

e A

SA

Pos

tula

te.

b.

�R

��

Yan

d S�

R��

X�Y�

.If

�S

��

X,t

hen

�R

ST

��

YX

Wby

th

e A

SA

Pos

tula

te.

Wh

at c

orre

spon

din

g p

arts

mu

st b

e co

ngr

uen

t in

ord

er t

o p

rove

th

at t

he

tria

ngl

esar

e co

ngr

uen

t b

y th

e A

SA

Pos

tula

te?

Wri

te t

he

tria

ngl

e co

ngr

uen

ce s

tate

men

t.

1.2.

3.

D�C�

�B�

C�;

W�Y�

�W�

Y�;

�A

BE

��

CB

D;

�C

DE

��

CB

A�

XY

W�

�Z

YW

;�

AB

E�

�C

BD

�W

XY

��

WZ

Y

4.5.

6.

B�D�

�D�

B�;

S�T�

�V�

T�;�

AC

B�

�E

;�

AD

B �

�C

BD

;�

RS

T�

�U

VT

�A

BC

��

CD

E�

AB

D�

�C

DB

AC

B

E

D

S

V

U

RT

D

AB

C

DC

EA

B

YW

X ZE

A

BD

C

RT

WY

SX

AC

B

DF

E

Exam

ple

Exam

ple

Exer

cises

Exer

cises

©G

lenc

oe/M

cGra

w-H

ill20

8G

lenc

oe G

eom

etry

AA

S Th

eore

mA

not

her

way

to

show

th

at t

wo

tria

ngl

es a

re c

ongr

uen

t is

th

e A

ngl

e-A

ngl

e-S

ide

(AA

S) T

heo

rem

.

AA

S T

heo

rem

If tw

o an

gles

and

a n

onin

clud

ed s

ide

of o

ne t

riang

le a

re c

ongr

uent

to

the

corr

espo

ndin

g tw

oan

gles

and

sid

e of

a s

econ

d tr

iang

le,

then

the

tw

o tr

iang

les

are

cong

ruen

t.

You

now

hav

e fi

ve w

ays

to s

how

th

at t

wo

tria

ngl

es a

re c

ongr

uen

t.•

defi

nit

ion

of

tria

ngl

e co

ngr

uen

ce•

AS

A P

ostu

late

•S

SS

Pos

tula

te•

AA

S T

heo

rem

•S

AS

Pos

tula

te

In t

he

dia

gram

,�B

CA

��

DC

A.W

hic

h s

ides

ar

e co

ngr

uen

t? W

hic

h a

dd

itio

nal

pai

r of

cor

resp

ond

ing

par

ts

nee

ds

to b

e co

ngr

uen

t fo

r th

e tr

ian

gles

to

be

con

gru

ent

by

the

AA

S P

ostu

late

?A �

C��

A�C�

by t

he

Ref

lexi

ve P

rope

rty

of c

ongr

uen

ce.T

he

con

gru

ent

angl

es c

ann

ot b

e �

1 an

d �

2,be

cau

se A �

C�w

ould

be

the

incl

ude

d si

de.

If �

B�

�D

,th

en �

AB

C�

�A

DC

by t

he

AA

S T

heo

rem

.

In E

xerc

ises

1 a

nd

2,d

raw

an

d l

abel

�A

BC

and

�D

EF

.In

dic

ate

wh

ich

ad

dit

ion

alp

air

of c

orre

spon

din

g p

arts

nee

ds

to b

e co

ngr

uen

t fo

r th

e tr

ian

gles

to

be

con

gru

ent

by

the

AA

S T

heo

rem

.

1.�

A�

�D

;�B

��

E2.

BC

�E

F;�

A�

�D

If B�

C��

E�F�

(or

if A�

C��

D�F�

),If

�C

��

F (

or

if �

B�

�E

),th

en �

AB

C�

�D

EF

by

the

then

�A

BC

��

DE

F by

the

A

AS

Th

eore

m.

AA

S T

heor

em.

3.W

rite

a f

low

pro

of.

Giv

en:�

S�

�U

;T �R�

bise

cts

�S

TU

.P

rove

:�S

RT

��

UR

T

Give

n

Give

n

RT

� R

T

Refl.

Pro

p. o

f �

Def.o

f � b

isec

tor

TR b

isec

ts �

STU

.

�SR

T �

�U

RT

�ST

R �

�U

TR

AAS

�SR

T �

�U

RT

CPCT

C�

S �

�U

S

RT

U

B

A

C

E

D

F

CA

B

FD

E

D

C1 2

A

B

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Pro

vin

g C

on

gru

ence

—A

SA

,AA

S

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Pro

vin

g C

on

gru

ence

—A

SA

,AA

S

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5

©G

lenc

oe/M

cGra

w-H

ill20

9G

lenc

oe G

eom

etry

Lesson 4-5

Wri

te a

flo

w p

roof

.

1.G

iven

:�

N�

�L

J �K��

M�K�

Pro

ve:

�J

KN

��

MK

L

Pro

of:

2.G

iven

:A�

B��

C�B�

�A

��

CD �

B�bi

sect

s �

AB

C.

Pro

ve:

A �D�

�C�

D�

Pro

of:

3.W

rite

a p

arag

raph

pro

of.

Giv

en:

D�E�

|| F�G�

�E

��

GP

rove

:�

DF

G�

�F

DE

Pro

of:

Sin

ce it

is g

iven

th

at D�

E�|| F�

G�,i

t fo

llow

s th

at �

ED

F�

�G

FD

,b

ecau

se a

lt.i

nt.

�ar

e �

.It

is g

iven

th

at �

E�

�G

.By

the

Ref

lexi

ve

Pro

per

ty, D�

F��

F�D�.S

o �

DF

G�

�F

DE

by A

AS

.

FG

DE

�A

� �

C

Give

nA

B �

CB

Give

nCP

CTC

AD

� C

D

DB

bis

ects

�A

BC

. Gi

ven

�A

BD

� �

CB

DAS

A

�A

BD

� �

CB

DDe

f. of

� b

isec

tor

AC

B D

�N

� �

LGi

ven

JK �

MK

Gi

ven

�JK

N �

�M

KL

Verti

cal �

are

�.

�JK

N �

�M

KL

AAS

N

J

M

KL

©G

lenc

oe/M

cGra

w-H

ill21

0G

lenc

oe G

eom

etry

1.W

rite

a f

low

pro

of.

Giv

en:

Sis

th

e m

idpo

int

of Q �

T�.

Q �R�

|| T�U�

Pro

ve:

�Q

SR

��

TS

US

amp

le p

roo

f:

2.W

rite

a p

arag

raph

pro

of.

Giv

en:

�D

��

FG �

E�bi

sect

s �

DE

F.

Pro

ve:

D �G�

�F�

G�

Pro

of:

Sin

ce it

is g

iven

th

at G�

E�b

isec

ts �

DE

F,�

DE

G�

�F

EG

by t

he

def

init

ion

of

an a

ng

le b

isec

tor.

It is

giv

en t

hat

�D

��

F.B

y th

e R

efle

xive

Pro

per

ty, G�

E��

G�E�

.So

�D

EG

��

FE

Gby

AA

S.T

her

efo

re

D�G�

�F�G�

by C

PC

TC

.

AR

CH

ITEC

TUR

EF

or E

xerc

ises

3 a

nd

4,u

se t

he

foll

owin

g in

form

atio

n.

An

arc

hit

ect

use

d th

e w

indo

w d

esig

n i

n t

he

diag

ram

wh

en r

emod

elin

g an

art

stu

dio.

A �B�

and

C�B�

each

mea

sure

3 f

eet.

3.S

upp

ose

Dis

th

e m

idpo

int

of A�

C�.D

eter

min

e w

het

her

�A

BD

��

CB

D.

Just

ify

you

r an

swer

.

Sin

ce D

is t

he

mid

po

int

of

A�C�

,A�D�

�C�

D�by

th

e d

efin

itio

n o

f m

idp

oin

t.A�

B��

C�B�

by t

he

def

init

ion

of

con

gru

ent

seg

men

ts.B

y th

e R

efle

xive

P

rop

erty

, B�D�

�B�

D�.S

o �

AB

D�

�C

BD

by S

SS

.

4.S

upp

ose

�A

��

C.D

eter

min

e w

het

her

�A

BD

��

CB

D.J

ust

ify

you

r an

swer

.

We

are

giv

en A�

B��

C�B�

and

�A

��

C.B�

D��

B�D�

by t

he

Ref

lexi

ve

Pro

per

ty.S

ince

SS

A c

ann

ot

be

use

d t

o p

rove

th

at t

rian

gle

s ar

e co

ng

ruen

t,w

e ca

nn

ot

say

wh

eth

er �

AB

D�

�C

BD

.

DB

AC

D

G

F

E

�Q

� �

T

Give

n

QR

|| TU

Gi

ven

Def.o

f mid

poin

t

Alt.

Int.

� a

re �

.

QS

� T

S S

is t

he

mid

po

int

of

QT.

�Q

SR �

�TS

UAS

A

�Q

SR �

�TS

UVe

rtica

l � a

re �

.

UQ

S

RT

Pra

ctic

e (

Ave

rag

e)

Pro

vin

g C

on

gru

ence

—A

SA

,AA

S

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5



Readin

g t

o L

earn

Math

em

ati

csP

rovi

ng

Co

ng

ruen

ce—

AS

A,A

AS

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5

©G

lenc

oe/M

cGra

w-H

ill21

1G

lenc

oe G

eom

etry

Lesson 4-5

Pre-

Act

ivit

yH

ow a

re c

ongr

uen

t tr

ian

gles

use

d i

n c

onst

ruct

ion

?R

ead

the

intr

odu

ctio

n t

o L

esso

n 4

-5 a

t th

e to

p of

pag

e 20

7 in

you

r te

xtbo

ok.

Wh

ich

of

the

tria

ngl

es i

n t

he

phot

ogra

ph i

n y

our

text

book

app

ear

to b

eco

ngru

ent?

Sam

ple

an

swer

:Th

e fo

ur

rig

ht

tria

ng

les

are

con

gru

ent

to e

ach

oth

er.T

he

two

ob

tuse

iso

scel

es t

rian

gle

s ar

e co

ng

ruen

tto

eac

h o

ther

.R

ead

ing

th

e Le

sso

n1.

Exp

lain

in

you

r ow

n w

ords

th

e di

ffer

ence

bet

wee

n h

ow t

he

AS

A P

ostu

late

an

d th

e A

AS

Th

eore

m a

re u

sed

to p

rove

th

at t

wo

tria

ngl

es a

re c

ongr

uen

t.S

amp

le a

nsw

er:

In A

SA

,yo

u u

se t

wo

pai

rs o

f co

ng

ruen

t an

gle

s an

d t

he

incl

ud

edco

ng

ruen

t si

des

.In

AA

S,y

ou

use

tw

o p

airs

of

con

gru

ent

ang

les

and

a p

air

of

no

nin

clu

ded

con

gru

ent

sid

es.

B,D

,E,G

,H2.

Whi

ch o

f th

e fo

llow

ing

cond

itio

ns a

re s

uffi

cien

t to

pro

ve t

hat

two

tria

ngle

s ar

e co

ngru

ent?

A.

Tw

o si

des

of o

ne

tria

ngl

e ar

e co

ngr

uen

t to

tw

o si

des

of t

he

oth

er t

rian

gle.

B.T

he

thre

e si

des

of o

ne

tria

ngl

es a

re c

ongr

uen

t to

th

e th

ree

side

s of

th

e ot

her

tri

angl

e.C

.T

he t

hree

ang

les

of o

ne t

rian

gle

are

cong

ruen

t to

the

thr

ee a

ngle

s of

the

oth

er t

rian

gle.

D.A

ll s

ix c

orre

spon

din

g pa

rts

of t

wo

tria

ngl

es a

re c

ongr

uen

t.E

.T

wo

angl

es a

nd

the

incl

ude

d si

de o

f 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.F.

Tw

o si

des

and

a n

onin

clu

ded

angl

e of

on

e tr

ian

gle

are

con

gru

ent

to t

wo

side

s an

d a

non

incl

ude

d an

gle

of t

he

oth

er t

rian

gle.

G.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.

H.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.I.

Tw

o an

gles

an

d a

non

incl

ude

d si

de o

f on

e tr

ian

gle

are

con

gru

ent

to t

wo

angl

es a

nd

an

onin

clu

ded

side

of

the

oth

er t

rian

gle.

3.D

eter

min

e w

het

her

you

hav

e en

ough

in

form

atio

n t

o pr

ove

that

th

e tw

o tr

ian

gles

in

eac

hfi

gure

are

con

gru

ent.

If s

o,w

rite

a c

ongr

uen

ce s

tate

men

t an

d n

ame

the

con

gru

ence

post

ula

te o

r th

eore

m t

hat

you

wou

ld u

se.I

f n

ot,w

rite

not

pos

sibl

e.

a.�

AE

B�

�D

EC

;A

AS

b.

Tis

th

e m

idpo

int

of R�

U�.

�R

ST

��

UV

T;

AS

A

Hel

pin

g Y

ou

Rem

emb

er4.

A g

ood

way

to

rem

embe

r m

athe

mat

ical

idea

s is

to

sum

mar

ize

them

in a

gen

eral

sta

tem

ent.

If y

ou w

ant

to p

rove

tri

angl

es c

ongr

uen

t by

usi

ng

thre

e pa

irs

of c

orre

spon

din

g pa

rts,

wh

at i

s a

good

way

to

rem

embe

r w

hic

h c

ombi

nat

ion

s of

par

ts w

ill

wor

k?S

amp

le a

nsw

er:A

t le

ast

on

e p

air

of

corr

esp

on

din

g p

arts

mu

st b

e si

des

.If

you

use

tw

o p

airs

of

sid

es a

nd

on

e p

air

of

ang

les,

the

ang

les

mu

st b

e th

ein

clu

ded

an

gle

s.If

yo

u u

se t

wo

pai

rs o

f an

gle

s an

d o

ne

pai

r o

f si

des

,th

en t

he

sid

es m

ust

bo

th b

e in

clu

ded

by

the

ang

les

or

mu

st b

oth

be

corr

esp

on

din

g n

on

incl

ud

ed s

ides

.

RS

T

U V

AD

CB

E

©G

lenc

oe/M

cGra

w-H

ill21

2G

lenc

oe G

eom

etry

Co

ng

ruen

t Tri

ang

les

in t

he

Co

ord

inat

e P

lan

eIf

you

kn

ow t

he

coor

din

ates

of

the

vert

ices

of

two

tria

ngl

es i

n t

he

coor

din

ate

plan

e,yo

u c

an o

ften

dec

ide

wh

eth

er t

he

two

tria

ngl

es a

re c

ongr

uen

t.T

her

em

ay b

e m

ore

than

on

e w

ay t

o do

th

is.

1.C

onsi

der

�A

BD

and

�C

DB

wh

ose

vert

ices

hav

e co

ordi

nat

es A

(0,0

),B

(2,5

),C

(9,5

),an

d D

(7,0

).B

rief

ly d

escr

ibe

how

you

can

use

wh

at y

oukn

ow a

bou

t co

ngr

uen

t tr

ian

gles

an

d th

e co

ordi

nat

e pl

ane

to s

how

th

at

�A

BD

��

CD

B.Y

ou m

ay w

ish

to

mak

e a

sket

ch t

o h

elp

get

you

sta

rted

.

Sam

ple

an

swer

:S

ho

w t

hat

th

e sl

op

es o

f A �

B �an

d C �

D �ar

eeq

ual

an

d t

hat

th

e sl

op

es o

f A �

D �an

d B �

C �ar

e eq

ual

.Co

ncl

ud

eth

at A �

B � �

C �D �

and

B �C �

�A �

D �.U

se t

he

ang

le r

elat

ion

ship

s fo

rp

aral

lel l

ines

an

d a

tra

nsv

ersa

l an

d t

he

fact

th

at B �

D �is

a c

om

-m

on

sid

e fo

r th

e tr

ian

gle

s to

co

ncl

ud

e th

at

�A

BD

��

CD

Bby

AS

A.

2.C

onsi

der

�P

QR

and

�K

LM

wh

ose

vert

ices

are

th

e fo

llow

ing

poin

ts.

P(1

,2)

Q(3

,6)

R(6

,5)

K(�

2,1)

L(�

6,3)

M(�

5,6)

Bri

efly

des

crib

e h

ow y

ou c

an s

how

th

at �

PQ

R�

�K

LM

.

Use

th

e D

ista

nce

Fo

rmu

la t

o f

ind

th

e le

ng

ths

of

the

sid

es o

fb

oth

tri

ang

les.

Co

ncl

ud

e th

at �

PQ

R�

�K

LM

by S

SS

.

3.If

you

kn

ow t

he

coor

din

ates

of

all

the

vert

ices

of

two

tria

ngl

es,i

s it

al

way

spo

ssib

le t

o te

ll w

het

her

th

e tr

ian

gles

are

con

gru

ent?

Exp

lain

.

Yes;

you

can

use

th

e D

ista

nce

Fo

rmu

la a

nd

SS

S.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-5

4-5



An

swer

s

Stu

dy G

uid

e a

nd I

nte

rven

tion

Iso

scel

es T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6

©G

lenc

oe/M

cGra

w-H

ill21

3G

lenc

oe G

eom

etry

Lesson 4-6

Pro

per

ties

of

Iso

scel

es T

rian

gle

sA

n i

sosc

eles

tri

angl

eh

as t

wo

con

gru

ent

side

s.T

he

angl

e fo

rmed

by

thes

e si

des

is c

alle

d th

e ve

rtex

an

gle.

Th

e ot

her

tw

o an

gles

are

cal

led

bas

e an

gles

.You

can

pro

ve a

th

eore

m a

nd

its

con

vers

e ab

out

isos

cele

s tr

ian

gles

.

•If

tw

o si

des

of a

tri

angl

e ar

e co

ngr

uen

t,th

en t

he

angl

es o

ppos

ite

thos

e si

des

are

con

gru

ent.

(Iso

scel

es T

rian

gle

Th

eore

m)

•If

tw

o an

gles

of

a tr

ian

gle

are

con

gru

ent,

then

th

e si

des

oppo

site

th

ose

angl

es a

re c

ongr

uen

t.If

A�B�

�C�

B�,

then

�A

��

C.

If �

A�

�C

, th

en A�

B��

C�B�

.

A

B

C

Fin

d x

.

BC

�B

A,s

o m

�A

�m

�C

.Is

os. T

riang

le T

heor

em

5x�

10 �

4x�

5S

ubst

itutio

n

x�

10 �

5S

ubtr

act

4xfr

om e

ach

side

.

x�

15A

dd 1

0 to

eac

h si

de.

B

AC( 4

x �

5) �

( 5x

� 1

0)�

Fin

d x

.

m�

S�

m�

T,s

oS

R�

TR

.C

onve

rse

of I

sos.

�T

hm.

3x�

13 �

2xS

ubst

itutio

n

3x�

2x�

13A

dd 1

3 to

eac

h si

de.

x�

13S

ubtr

act

2xfr

om e

ach

side

.

RT

S 3x �

13

2x

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exer

cises

Exer

cises

Fin

d x

.

1.35

2.12

3.15

4.12

5.20

6.36

7.W

rite

a t

wo-

colu

mn

pro

of.

Giv

en:�

1 �

�2

Pro

ve:A �

B��

C�B�

Sta

tem

ents

Rea

son

s

1.�

1 �

�2

1.G

iven

2.�

2 �

�3

2.V

erti

cal a

ng

les

are

con

gru

ent.

3.�

1 �

�3

3.Tr

ansi

tive

Pro

per

ty o

f �

4.A�

B��

C�B�

4.If

tw

o a

ng

les

of

a tr

ian

gle

are

�,t

hen

th

e si

des

op

po

site

th

e an

gle

s ar

e �

.

B

AC

D

E

13

2

RS

T 3x�

x�D

BG

L3x

�

30�

D TQP

K

( 6x

� 6

) �2x

�

W

YZ

3x�

S

V

T3x

� 6

2x �

6R

P

Q2x

�40

�

©G

lenc

oe/M

cGra

w-H

ill21

4G

lenc

oe G

eom

etry

Pro

per

ties

of

Equ

ilate

ral T

rian

gle

sA

n e

qu

ilat

eral

tri

angl

eh

as t

hre

e co

ngr

uen

tsi

des.

Th

e Is

osce

les

Tri

angl

e T

heo

rem

can

be

use

d to

pro

ve t

wo

prop

erti

es o

f eq

uil

ater

altr

ian

gles

.

1.A

tria

ngle

is e

quila

tera

l if

and

only

if it

is e

quia

ngul

ar.

2.E

ach

angl

e of

an

equi

late

ral t

riang

le m

easu

res

60°.

Pro

ve t

hat

if

a li

ne

is p

aral

lel

to o

ne

sid

e of

an

eq

uil

ater

al t

rian

gle,

then

it

form

s an

oth

er e

qu

ilat

eral

tr

ian

gle.

Pro

of:

Sta

tem

ents

Rea

son

s

1.�

AB

Cis

equ

ilat

eral

;P�Q�

|| B�C�

.1.

Giv

en2.

m�

A�

m�

B�

m�

C�

602.

Eac

h �

of a

n e

quil

ater

al �

mea

sure

s 60

°.3.

�1

��

B,�

2 �

�C

3.If

||li

nes

,th

en c

orre

s.�

s ar

e �

.4.

m�

1 �

60,m

�2

�60

4.S

ubs

titu

tion

5.�

AP

Qis

equ

ilat

eral

.5.

If a

�is

equ

ian

gula

r,th

en i

t is

equ

ilat

eral

.

Fin

d x

.

1.10

2.5

3.10

4.10

5.12

6.15

7.W

rite

a t

wo-

colu

mn

pro

of.

Giv

en:�

AB

Cis

equ

ilat

eral

;�1

��

2.P

rove

:�A

DB

��

CD

B

Pro

of:

Sta

tem

ents

Rea

son

s

1.�

AB

Cis

eq

uila

tera

l.1.

Giv

en2.

A�B�

�C�

B�;

�A

��

C2.

An

eq

uila

tera

l �h

as �

sid

es a

nd

�an

gle

s.3.

�1

��

23.

Giv

en4.

�A

BD

��

CB

D4.

AS

A P

ost

ula

te5.

�A

DB

��

CD

B5.

CP

CT

CA D C

B1 2

R O

HM

60�

4x�

X

ZY

4x �

4

3x �

860

�

PQ

LV

R60

�

4x40

L N M

K

�K

LM is

equ

ilate

ral.

3x�

G

JH

6x �

55 x

D

FE

6x�

A

B

PQ

C

12

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Iso

scel

es T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6

Exam

ple

Exam

ple

Exer

cises

Exer

cises



Skil

ls P

ract

ice

Iso

scel

es T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6

©G

lenc

oe/M

cGra

w-H

ill21

5G

lenc

oe G

eom

etry

Lesson 4-6

Ref

er t

o th

e fi

gure

.

1.If

A�C�

�A�

D�,n

ame

two

con

gru

ent

angl

es.

�A

CD

��

CD

A

2.If

B�E�

�B�

C�,n

ame

two

con

gru

ent

angl

es.

�B

EC

��

BC

E

3.If

�E

BA

��

EA

B,n

ame

two

con

gru

ent

segm

ents

.

E�B�

�E�

A�

4.If

�C

ED

��

CD

E,n

ame

two

con

gru

ent

segm

ents

.

C�E�

�C�

D�

�A

BF

is i

sosc

eles

,�C

DF

is e

qu

ilat

eral

,an

d m

�A

FD

�15

0.F

ind

eac

h m

easu

re.

5.m

�C

FD

606.

m�

AF

B55

7.m

�A

BF

708.

m�

A55

In t

he

figu

re,P�

L��

R�L�

and

L�R�

�B�

R�.

9.If

m�

RL

P�

100,

fin

d m

�B

RL

.20

10.I

f m

�L

PR

�34

,fin

d m

�B

.68

11.W

rite

a t

wo-

colu

mn

pro

of.

Giv

en:

C�D�

�C �

G�D�

E��

G�F�

Pro

ve:

C�E�

�C�

F�

Pro

of:

Sta

tem

ents

Rea

son

s

1.C�

D��

C�G�

1.G

iven

2.�

D�

�G

2.If

2 s

ides

of

a �

are

�,t

hen

th

e �

op

po

site

tho

se s

ides

are

�.

3.D�

E��

G�F�

3.G

iven

4.�

CD

E�

�C

GF

4.S

AS

5.C�

E��

C�F�

5.C

PC

TC

D E F G

CRP

BL

D

C F

B

35�

AE

D

C

B

AE

©G

lenc

oe/M

cGra

w-H

ill21

6G

lenc

oe G

eom

etry

Ref

er t

o th

e fi

gure

.

1.If

R�V�

�R�

T�,n

ame

two

con

gru

ent

angl

es.

�R

TV

��

RV

T

2.If

R�S�

�S�

V�,n

ame

two

con

gru

ent

angl

es.

�S

VR

��

SR

V

3.If

�S

RT

��

ST

R,n

ame

two

con

gru

ent

segm

ents

.S�

T��

S�R�

4.If

�S

TV

��

SV

T,n

ame

two

con

gru

ent

segm

ents

.S�

T��

S�V�

Tri

angl

es G

HM

and

HJ

Mar

e is

osce

les,

wit

h G�

H��

M�H�

and

H�J�

�M�

J�.T

rian

gle

KL

Mis

eq

uil

ater

al,a

nd

m�

HM

K�

50.

Fin

d e

ach

mea

sure

.

5.m

�K

ML

606.

m�

HM

G70

7.m

�G

HM

40

8.If

m�

HJ

M�

145,

fin

d m

�M

HJ

.17

.5

9.If

m�

G�

67,f

ind

m�

GH

M.

46

10.W

rite

a t

wo-

colu

mn

pro

of.

Giv

en:

D�E�

|| B�C�

�1

��

2P

rove

:A �

B��

A�C�

Pro

of:

Sta

tem

ents

Rea

son

s

1.D�

E�|| B�

C�1.

Giv

en

2.�

1 �

�4

2.C

orr

.�ar

e �

.�

2 �

�3

3.�

1 �

�2

3.G

iven

4.�

3 �

�4

4.C

on

gru

ence

of

�is

tra

nsi

tive

.

5.A�

B��

A�C�

5.If

2 �

of

a �

are

�,t

hen

th

e si

des

op

po

site

th

ose

�ar

e �

.

11.S

POR

TSA

pen

nan

t fo

r th

e sp

orts

tea

ms

at L

inco

ln H

igh

S

choo

l is

in

th

e sh

ape

of a

n i

sosc

eles

tri

angl

e.If

th

e m

easu

re

of t

he

vert

ex a

ngl

e is

18,

fin

d th

e m

easu

re o

f ea

ch b

ase

angl

e.81

,81

Linc

oln

Haw

ks

E DBC

A12

3 4

GMLK

J

H

UR

TV

S

Pra

ctic

e (

Ave

rag

e)

Iso

scel

es T

rian

gle

s

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6



An

swer

s

Readin

g t

o L

earn

Math

em

ati

csIs

osc

eles

Tri

ang

les

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6

©G

lenc

oe/M

cGra

w-H

ill21

7G

lenc

oe G

eom

etry

Lesson 4-6

Pre-

Act

ivit

yH

ow a

re t

rian

gles

use

d i

n a

rt?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-6

at

the

top

of p

age

216

in y

our

text

book

.

•W

hy

do y

ou t

hin

k th

at i

sosc

eles

an

d eq

uil

ater

al t

rian

gles

are

use

d m

ore

ofte

n t

han

sca

len

e tr

ian

gles

in

art

?S

amp

le a

nsw

er:T

hei

rsy

mm

etry

is p

leas

ing

to

th

e ey

e.•

Wh

y m

igh

t is

osce

les

righ

t tr

ian

gles

be

use

d in

art

?S

amp

le a

nsw

er:

Two

co

ng

ruen

t is

osc

eles

rig

ht

tria

ng

les

can

be

pla

ced

tog

eth

er t

o f

orm

a s

qu

are.

Rea

din

g t

he

Less

on

1.R

efer

to

the

figu

re.

a.W

hat

kin

d of

tri

angl

e is

�Q

RS

?is

osc

eles

b.

Nam

e th

e le

gs o

f �

QR

S.

Q�S�

,R�S�

c.N

ame

the

base

of

�Q

RS

.Q�

R�d

.N

ame

the

vert

ex a

ngl

e of

�Q

RS

.�

Se.

Nam

e th

e ba

se a

ngl

es o

f �

QR

S.

�Q

,�R

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.If

a t

rian

gle

has

th

ree

con

gru

ent

side

s,th

en i

t h

as t

hre

e co

ngr

uen

t an

gles

.al

way

sb

.If

a t

rian

gle

is i

sosc

eles

,th

en i

t is

equ

ilat

eral

.so

met

imes

c.If

a r

igh

t tr

ian

gle

is i

sosc

eles

,th

en i

t is

equ

ilat

eral

.n

ever

d.

Th

e la

rges

t an

gle

of a

n i

sosc

eles

tri

angl

e is

obt

use

.so

met

imes

e.If

a r

igh

t tr

ian

gle

has

a 4

5°an

gle,

then

it

is i

sosc

eles

.al

way

sf.

If a

n i

sosc

eles

tri

angl

e h

as t

hre

e ac

ute

an

gles

,th

en i

t is

equ

ilat

eral

.so

met

imes

g.T

he

vert

ex a

ngl

e of

an

iso

scel

es t

rian

gle

is t

he

larg

est

angl

e of

th

e tr

ian

gle.

som

etim

es3.

Giv

e th

e m

easu

res

of t

he

thre

e an

gles

of

each

tri

angl

e.

a.an

equ

ilat

eral

tri

angl

e60

,60,

60b

.an

iso

scel

es r

igh

t tr

ian

gle

45,4

5,90

c.an

iso

scel

es t

rian

gle

in w

hic

h t

he

mea

sure

of

the

vert

ex a

ngl

e is

70

70,5

5,55

d.

an i

sosc

eles

tri

angl

e in

wh

ich

th

e m

easu

re o

f a

base

an

gle

is 7

070

,70,

40e.

an i

sosc

eles

tri

angl

e in

wh

ich

th

e m

easu

re o

f th

e ve

rtex

an

gle

is t

wic

e th

e m

easu

re o

fon

e of

th

e ba

se a

ngl

es90

,45,

45

Hel

pin

g Y

ou

Rem

emb

er4.

If a

th

eore

m a

nd

its

con

vers

e ar

e bo

th t

rue,

you

can

oft

en r

emem

ber

them

mos

t ea

sily

by

com

bini

ng t

hem

into

an

“if-

and-

only

-if”

stat

emen

t.W

rite

suc

h a

stat

emen

t fo

r th

e Is

osce

les

Tri

angl

e T

heo

rem

an

d it

s co

nve

rse.

Sam

ple

an

swer

:Tw

o s

ides

of

a tr

ian

gle

are

con

gru

ent

if a

nd

on

ly if

th

e an

gle

s o

pp

osi

te t

ho

se s

ides

are

co

ng

ruen

t.

R Q

S

©G

lenc

oe/M

cGra

w-H

ill21

8G

lenc

oe G

eom

etry

Tria

ng

le C

hal

len

ges

Som

e pr

oble

ms

incl

ude

dia

gram

s.If

you

are

not

su

re h

ow t

o so

lve

the

prob

lem

,beg

in b

y u

sin

g th

e gi

ven

in

form

atio

n.F

ind

the

mea

sure

s of

as

man

yan

gles

as

you

can

,wri

tin

g ea

ch m

easu

re o

n t

he

diag

ram

.Th

is m

ay g

ive

you

mor

e cl

ues

to

the

solu

tion

.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-6

4-6

1.G

iven

:B

E�

BF

, �B

FG

� �

BE

F �

�B

ED

, m�

BF

E �

82 a

nd

AB

FG

and

BC

DE

each

hav

eop

posi

te s

ides

par

alle

l an

dco

ngr

uen

t.F

ind

m�

AB

C.

148

3.G

iven

:m

�U

ZY

�90

,m�

ZW

X�

45,

�Y

ZU

��

VW

X,U

VX

Yis

asq

uar

e (a

ll s

ides

con

gru

ent,

all

angl

es r

igh

t an

gles

).F

ind

m�

WZ

Y.

45

2.G

iven

:A

C�

AD

,an

d A �

B ��

B �D �

,m

�D

AC

�44

an

dC �

E �bi

sect

s �

AC

D.

Fin

d m

�D

EC

.78

4.G

iven

:m

�N

�12

0,J �N �

�M �

N �,

�J

NM

��

KL

M.

Fin

d m

�J

KM

.15

J

K

L

MN

A

DC

BE

UV

W

XY

Z

A

GD

FE

CB



Stu

dy G

uid

e a

nd I

nte

rven

tion

Tria

ng

les

and

Co

ord

inat

e P

roo

f

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7

©G

lenc

oe/M

cGra

w-H

ill21

9G

lenc

oe G

eom

etry

Lesson 4-7

Posi

tio

n a

nd

Lab

el T

rian

gle

sA

coo

rdin

ate

proo

f u

ses

poin

ts,d

ista

nce

s,an

d sl

opes

to

prov

e ge

omet

ric

prop

erti

es.T

he

firs

t st

ep i

n w

riti

ng

a co

ordi

nat

e pr

oof

is t

o pl

ace

a fi

gure

on

the

coor

din

ate

plan

e an

d la

bel

the

vert

ices

.Use

th

e fo

llow

ing

guid

elin

es.

1.U

se t

he o

rigin

as

a ve

rtex

or

cent

er o

f th

e fig

ure.

2.P

lace

at

leas

t on

e si

de o

f th

e po

lygo

n on

an

axis

.3.

Kee

p th

e fig

ure

in t

he f

irst

quad

rant

if p

ossi

ble.

4.U

se c

oord

inat

es t

hat

mak

e th

e co

mpu

tatio

ns a

s si

mpl

e as

pos

sibl

e.

Pos

itio

n a

n e

qu

ilat

eral

tri

angl

e on

th

e co

ord

inat

e p

lan

e so

th

at i

ts s

ides

are

au

nit

s lo

ng

and

on

e si

de

is o

n t

he

pos

itiv

e x-

axis

.S

tart

wit

h R

(0,0

).If

RT

is a

,th

en a

not

her

ver

tex

is T

(a,0

).

For

ver

tex

S,t

he

x-co

ordi

nat

e is

�a 2� .U

se b

for

the

y-co

ordi

nat

e,

so t

he

vert

ex i

s S��a 2� ,

b �.

Fin

d t

he

mis

sin

g co

ord

inat

es o

f ea

ch t

rian

gle.

1.2.

3.

C(p

,q)

T(2

a,2a

)E

(�2g

,0);

F(0

,b)

Pos

itio

n a

nd

lab

el e

ach

tri

angl

e on

th

e co

ord

inat

e p

lan

e.

4.is

osce

les

tria

ngl

e 5.

isos

cele

s ri

ght

�D

EF

6.eq

uila

tera

l tr

iang

le �

EQ

I�

RS

T w

ith

bas

e R �

S�w

ith

leg

s e

un

its

lon

gw

ith

ver

tex

Q(0

,a)

and

4au

nit

s lo

ng

side

s 2b

un

its

lon

g

x

y

I(b,

0)

E( –

b, 0

)

Q( 0

, a)

x

y

E( e

, 0)

F( e

, e)

D( 0

, 0)

x

yT

( 2a,

b)

R( 0

, 0)

S( 4

a, 0

)

Sam

ple

an

swer

sar

e g

iven

.

x

y

G( 2

g, 0

)

F( ?

, b)

E( ?

, ?)

x

y

S( 2

a, 0

)

T( ?

, ?)

R( 0

, 0)

x

y

B( 2

p, 0

)

C( ?

, q)

A( 0

, 0)

x

y

T( a

, 0)

R( 0

, 0)S

�a – 2, b�

Exer

cises

Exer

cises

Exam

ple

Exam

ple

©G

lenc

oe/M

cGra

w-H

ill22

0G

lenc

oe G

eom

etry

Wri

te C

oo

rdin

ate

Pro

ofs

Coo

rdin

ate

proo

fs c

an b

e u

sed

to p

rove

th

eore

ms

and

tove

rify

pro

pert

ies.

Man

y co

ordi

nat

e pr

oofs

use

th

e D

ista

nce

For

mu

la,S

lope

For

mu

la,o

rM

idpo

int

Th

eore

m.

Pro

ve t

hat

a s

egm

ent

from

th

e ve

rtex

an

gle

of a

n i

sosc

eles

tri

angl

e to

th

e m

idp

oin

t of

th

e b

ase

is p

erp

end

icu

lar

to t

he

bas

e.F

irst

,pos

itio

n a

nd

labe

l an

iso

scel

es t

rian

gle

on t

he

coor

din

ate

plan

e.O

ne

way

is

to u

se T

(a,0

),R

(�a,

0),a

nd

S(0

,c).

Th

en U

(0,0

) is

th

e m

idpo

int

of R �

T�.

Giv

en:I

sosc

eles

�R

ST

;Uis

th

e m

idpo

int

of b

ase

R �T�

.P

rove

:S �U�

⊥R�

T�

Pro

of:

Uis

th

e m

idpo

int

of R �

T�so

th

e co

ordi

nat

es o

f U

are ��

a 2�a

�,�

0� 2

0�

��(0

,0).

Th

us

S�U�

lies

on

the

y-ax

is,a

nd

�R

ST

was

pla

ced

so R�

T�li

es o

n t

he

x-ax

is.T

he

axes

are

per

pen

dicu

lar,

so

S �U�

⊥R�

T�.

Pro

ve t

hat

th

e se

gmen

ts j

oin

ing

the

mid

poi

nts

of

the

sid

es o

f a

righ

t tr

ian

gle

form

a ri

ght

tria

ngl

e.

Sam

ple

an

swer

:P

osi

tio

n a

nd

lab

el r

igh

t �

AB

Cw

ith

th

e co

ord

inat

es

A(0

,0),

B(0

,2b

),an

d C

(2a,

0).

Th

e m

idp

oin

t P

of

BC

is ��0

� 22a �

,�2b

2�0

��

(a,b

).

Th

e m

idp

oin

t Q

of

AC

is ��0

� 22a �

,�0

� 20

��

(a,0

).

Th

e m

idp

oin

t R

of

AB

is ��0

� 20

�,�

0� 2

2b ��

(0,b

).

Th

e sl

op

e o

f R�

P�is

�b a� �

b 0�

��0 a�

�0,

so t

he

seg

men

t is

ho

rizo

nta

l.

Th

e sl

op

e o

f P�

Q�is

�b a��

a0�

��b 0� ,

wh

ich

is u

nd

efin

ed,s

o t

he

seg

men

t is

ver

tica

l.

�R

PQ

is a

rig

ht

ang

le b

ecau

se a

ny h

ori

zon

tal l

ine

is p

erp

end

icu

lar

to a

nyve

rtic

al li

ne.

�P

RQ

has

a r

igh

t an

gle

,so

�P

RQ

is a

rig

ht

tria

ng

le.

x

y

C( 2

a, 0

)

B( 0

, 2b)

P Q

R

A( 0

, 0)

x

y

T( a

, 0)

U( 0

, 0)

R( –

a, 0

)

S( 0

, c)

Stu

dy G

uid

e a

nd I

nte

rven

tion

(con

tinued

)

Tria

ng

les

and

Co

ord

inat

e P

roo

f

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7

Exam

ple

Exam

ple

Exer

cises

Exer

cises



An

swer

s

Skil

ls P

ract

ice

Tria

ng

les

and

Co

ord

inat

e P

roo

f

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7

©G

lenc

oe/M

cGra

w-H

ill22

1G

lenc

oe G

eom

etry

Lesson 4-7

Pos

itio

n a

nd

lab

el e

ach

tri

angl

e on

th

e co

ord

inat

e p

lan

e.

1.ri

ght

�F

GH

wit

h l

egs

2.is

osce

les

�K

LP

wit

h

3.is

osce

les

�A

ND

wit

ha

un

its

and

bu

nit

sba

se K �

P�6b

un

its

lon

gba

se A�

D�5a

lon

g

Fin

d t

he

mis

sin

g co

ord

inat

es o

f ea

ch t

rian

gle.

4.5.

6.

A(0

,2a)

Z(b

,c)

M(0

,c)

7.8.

9.

Q(4

a,0)

R��7 2� b

,c�

T(0

,b)

10.W

rite

a c

oord

inat

e pr

oof

to p

rove

th

at i

n a

n i

sosc

eles

rig

ht

tria

ngl

e,th

e se

gmen

t fr

omth

e ve

rtex

of

the

righ

t an

gle

to t

he

mid

poin

t of

th

e h

ypot

enu

se i

s pe

rpen

dicu

lar

to t

he

hyp

oten

use

.

Giv

en:

isos

cele

s ri

ght

�A

BC

wit

h �

AB

Cth

e ri

ght

angl

e an

d M

the

mid

poin

t of

A �C�

Pro

ve:

B�M�

⊥A�

C�

Pro

of:

Th

e M

idp

oin

t F

orm

ula

sh

ow

s th

at t

he

coo

rdin

ates

of

Mar

e ��0

� 22a �

,�2a

2�0

��o

r (a

,a).

Th

e sl

op

e o

f A�

C�is

�2 0a ��

20 a�

��

1.T

he

slo

pe

of

B�M�

is �a a

� �0 0

��

1.T

he

pro

du

ct

of

the

slo

pes

is �

1,so

B�M�

⊥A�

C�.

x

y

C( 2

a, 0

)

A( 0

, 2a)

M

B( 0

, 0)

x

y

U( a

, 0)

T( ?

, ?)

S( –

a, 0

)x

y

P( 7

b, 0

)

R( ?

, ?)

N( 0

, 0)

x

y

Q( ?

, ?)

R( 2

a, b

)

P( 0

, 0)

x

y

N( 3

b, 0

)

M( ?

, ?)

O( 0

, 0)

x

y

Y( 2

b, 0

)

Z( ?

, ?)

X( 0

, 0)

x

y

B( 2

a, 0

)

A( 0

, ?)

C( 0

, 0)

x

yN

�5 – 2a, b

�

A( 0

, 0)

D( 5

a, 0

)x

yL(

3b, c

)

K( 0

, 0)

P( 6

b, 0

)x

y F( 0

, a)

G( 0

, 0)

H( b

, 0)

Sam

ple

an

swer

sar

e g

iven

.

©G

lenc

oe/M

cGra

w-H

ill22

2G

lenc

oe G

eom

etry

Pos

itio

n a

nd

lab

el e

ach

tri

angl

e on

th

e co

ord

inat

e p

lan

e.

1.eq

uil

ater

al �

SW

Yw

ith

2.

isos

cele

s �

BL

Pw

ith

3.

isos

cele

s ri

ght

�D

GJ

side

s �1 4� a

lon

gba

se B�

L�3b

un

its

lon

gw

ith

hyp

oten

use

D�J�

and

legs

2a

un

its

lon

g

Fin

d t

he

mis

sin

g co

ord

inat

es o

f ea

ch t

rian

gle.

4.5.

6.

S��1 6� b

,c�

C(3

a,0)

,E(0

,c)

M(0

,c),

N(�

2b,0

)

NEI

GH

BO

RH

OO

DS

For

Exe

rcis

es 7

an

d 8

,use

th

e fo

llow

ing

info

rmat

ion

.K

arin

a li

ves

6 m

iles

eas

t an

d 4

mil

es n

orth

of

her

hig

h s

choo

l.A

fter

sch

ool

she

wor

ks p

art

tim

e at

the

mal

l in

a m

usic

sto

re.T

he m

all

is 2

mil

es w

est

and

3 m

iles

nor

th o

f th

e sc

hool

.

7.W

rite

a c

oord

inat

e pr

oof

to p

rove

th

at K

arin

a’s

hig

h s

choo

l,h

er h

ome,

and

the

mal

l ar

eat

th

e ve

rtic

es o

f a

righ

t tr

ian

gle.

Giv

en:

�S

KM

Pro

ve:

�S

KM

is a

rig

ht

tria

ngl

e.

Pro

of:

Slo

pe

of

SK

��4 6

� �0 0

�o

r �2 3�

Slo

pe

of

SM

�� 3 2� �

0 0�

or

��3 2�

Sin

ce t

he

slo

pe

of

S�M�

is t

he

neg

ativ

e re

cip

roca

l of

the

slo

pe

of

S�K�

,S�M�

⊥S�

K�.T

her

efo

re,�

SK

Mis

rig

ht

tria

ng

le.

8.F

ind

the

dist

ance

bet

wee

n t

he

mal

l an

d K

arin

a’s

hom

e.

KM

��

(�2

��

6)2

��

(3 �

�4)

2�

��

64 �

�1�

��

65�o

r �

8.1

mile

s

x

y S( 0

, 0)

K( 6

, 4)

M( –

2, 3

)

x

y

P( 2

b, 0

)

M( 0

, ?)

N( ?

, 0)

x

y

C( ?

, 0)

E( 0

, ?)

B( –

3a, 0

)x

yS

( ?, ?

)

J(0,

0)

R�1 – 3b,

0�

x

y D( 0

, 2a)

G( 0

, 0)

J(2a

, 0)

x

yP

�3 – 2b, c

�

B( 0

, 0)

L(3b

, 0)

x

yY

�1 – 8a, b

�

W�1 – 4a,

0�

S( 0

, 0)

Sam

ple

an

swer

sar

e g

iven

.

Pra

ctic

e (

Ave

rag

e)

Tria

ng

les

and

Co

ord

inat

e P

roo

f

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7



Readin

g t

o L

earn

Math

em

ati

csTr

ian

gle

s an

d C

oo

rdin

ate

Pro

of

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7

©G

lenc

oe/M

cGra

w-H

ill22

3G

lenc

oe G

eom

etry

Lesson 4-7

Pre-

Act

ivit

yH

ow c

an t

he

coor

din

ate

pla

ne

be

use

ful

in p

roof

s?

Rea

d th

e in

trod

uct

ion

to

Les

son

4-7

at

the

top

of p

age

222

in y

our

text

book

.

Fro

m t

he

coor

din

ates

of

A,B

,an

d C

in t

he

draw

ing

in y

our

text

book

,wh

atdo

you

kn

ow a

bou

t �

AB

C?

Sam

ple

an

swer

:�

AB

Cis

iso

scel

esw

ith

�C

as t

he

vert

ex a

ng

le.

Rea

din

g t

he

Less

on

1.F

ind

the

mis

sin

g co

ordi

nat

es o

f ea

ch t

rian

gle.

a.b

.

R(0

,b),

S(0

,0),

T�a,

�b 2� �D

(0,0

),E

(0,a

),F

(a,a

)

2.R

efer

to

the

figu

re.

a.F

ind

the

slop

e of

S�R�

and

the

slop

e of

S�T�

.1;

�1

b.

Fin

d th

e pr

odu

ct o

f th

e sl

opes

of

S�R�

and

S�T�

.Wh

at

does

th

is t

ell

you

abo

ut

S �R�

and

S�T�

?�

1;S�

R�⊥

S�T�

c.W

hat

doe

s yo

ur

answ

er f

rom

par

t b

tell

you

abo

ut

�R

ST

?S

amp

le a

nsw

er:

�R

ST

is a

rig

ht

tria

ng

le w

ith

�S

as t

he

rig

ht

ang

le.

d.

Fin

d S

Ran

d S

T.W

hat

doe

s th

is t

ell

you

abo

ut

S�R�

and

S�T�

?

SR

��

2a2

�o

r a�

2�;S

T�

�2a

2�

or

a�2�;

S�R�

�S�

T�e.

Wh

at d

oes

you

r an

swer

fro

m p

art

d te

ll y

ou a

bou

t �

RS

T?

Sam

ple

an

swer

:�

RS

Tis

iso

scel

es w

ith

�R

ST

as t

he

vert

ex a

ng

le.

f.C

ombi

ne y

our

answ

ers

from

par

ts c

and

e t

o de

scri

be �

RS

Tas

com

plet

ely

as p

ossi

ble.

Sam

ple

an

swer

:�

RS

Tis

an

iso

scel

es r

igh

t tr

ian

gle

.�R

ST

is t

he

rig

ht

ang

le a

nd

is a

lso

th

e ve

rtex

an

gle

.g.

Fin

d m

�S

RT

and

m�

ST

R.

45;

45h

.F

ind

m�

OS

Ran

d m

�O

ST

.45

;45

Hel

pin

g Y

ou

Rem

emb

er

3.M

any

stu

den

ts f

ind

it e

asie

r to

rem

embe

r m

ath

emat

ical

for

mu

las

if t

hey

can

pu

t th

emin

to w

ords

in

a c

ompa

ct w

ay.H

ow c

an y

ou u

se t

his

app

roac

h t

o re

mem

ber

the

slop

e an

dm

idpo

int

form

ula

s ea

sily

?S

amp

le a

nsw

er:

Slo

pe

Fo

rmu

la:

chan

ge

in y

over

ch

ang

e in

x;

Mid

po

int

Fo

rmu

la:

aver

age

of

x-co

ord

inat

es,a

vera

ge

of

y-co

ord

inat

es

x

y S( 0

, a)

R( –

a, 0

)T

( a, 0

)O

( 0, 0

)

x

y

F( ?

, ?)

E( ?

, a)

D( ?

, ?)

x

y

T( a

, ?)

R( ?

, b)

S( ?

, ?)

©G

lenc

oe/M

cGra

w-H

ill22

4G

lenc

oe G

eom

etry

How

Man

y Tr

ian

gle

s?E

ach

pu

zzle

bel

ow c

onta

ins

man

y tr

ian

gles

.Cou

nt

them

car

efu

lly.

Som

e tr

ian

gles

ove

rlap

oth

er t

rian

gles

.

How

man

y tr

ian

gles

are

th

ere

in e

ach

fig

ure

?

1.8

2.40

3.35

4.5

5.13

6.27

How

man

y tr

ian

gles

can

you

for

m b

y jo

inin

g p

oin

ts o

n e

ach

cir

cle?

L

ist

the

vert

ices

of

each

tri

angl

e.

7.8.

4;A

BC

,AB

D,A

CD

,BC

D10

;E

FG

,EF

H,E

FI,

EG

H,E

HI,

FG

H,

FG

I,F

HI,

EG

I,G

HI

8.9.

20;

JKL

,JK

M,J

KN

,JK

O,J

LM

,JL

N,J

LO

,JM

N,J

MO

,JN

O,K

LM

,K

LN

,KL

O,K

MN

,KM

O,K

NO

,L

MN

,LM

O,L

NO

,MN

O

35;

PQ

R,P

QS

,PQ

T,P

QU

,PQ

V,

PR

S,P

RT

,PR

U,P

RV

,PS

T,P

SU

,P

SV

,PT

U,P

TV

,PU

V,Q

RS

,QR

T,

QR

U,Q

RV

,QS

T,Q

SU

,QS

V,Q

TU

,Q

TV

,QU

V,R

ST

,RS

U,R

SV

,RT

U,

RT

V,R

UV,

ST

U,S

TV,

SU

V,T

UV

QR

P

U

S

TV

JK

O

L

MN

EF

I

GH

B

C

DA

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

AT

E__

____

____

__P

ER

IOD

____

_

4-7

4-7



Chapter 4 Assessment Answer Key Form 1 Form 2APage 225 Page 226 Page 227


An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

C

D

A

C

B

D

B

D

9.

10.

11.

12.

13.

B:

C

A

C

A

B

isosceles

1.

2.

3.

4.

5.

6.

7.

8.

C

C

A

D

B

D

D

A


Chapter 4 Assessment Answer KeyForm 2A (continued) Form 2BPage 228 Page 229 Page 230

9.

10.

11.

12.

13.

B:

B

C

B

A

A

A(0, 0), C(�a, a)

1.

2.

3.

4.

5.

6.

7.

8.

D

B

D

A

C

B

A

A

9.

10.

11.

12.

13.

B:

C

C

D

A

D

�2


Chapter 4 Assessment Answer KeyForm 2CPage 231 Page 232

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

acute scalene

x � 3, AB �

BC � AC � 24

EF � FG � 4�2�,EG � 8, isosceles

70

140

50

�DFG � �BAC,�D � �B, �F ��A, �G � �C

AB � A�B� ��29�,

BC � B�C� � �10�,AC � A�C� � �17�

9.

10.

11.

12.

13.

14.

B:

SAS

Isosceles �Theorem, AAS

45

x � 4

P��b2

�, �b2

��, C�P� ⊥ A�B�

�XYZ � �MNO

x

yC(b–

2, c)

B(b, 0)A


Chapter 4 Assessment Answer KeyForm 2DPage 233 Page 234

1.

2.

3.

4.

5.

6.

7.

8.

9.

obtuse isosceles

x � 5, AB �

BC � 30, AC � 40

EF � FG � 5,

EG � 5�2�,isosceles

110

70

30

�ABC � �FDE,�A � �F, �B ��D, �C � �E

JK � J�K � � �10�,

JL � J �L� � �29�,KL � K �L� � �37�

SAS

10.

11.

12.

13.

14.

B:

Isosceles �Theorem, AAS

50

x � 6

M ��a2

�, 0�, N�0, �b2

��,slopes: M�N� � ��

ba

�

and B�C� � ��ba

�

�ABC � �DEF

x

y

M(3a, 0)

L(1.5a, b)

K(0, 0)


Chapter 4 Assessment Answer KeyForm 3Page 235 Page 236

An

swer

s

1.

2.

3.

4.

5.

6.

7.

8.

x � 4, AB � 78,BC � 78,AC � 100

AB � 5, BC � 10,

AC � 3�5�;scalene obtuse

20

90

40

AB � A�B�� 26�,

BC � B�C� � 3�2�,AC � A�C� � 2�5�

GH � JK � 2�5�,

IG � LJ � 2�5�,IH � LK � 2�2�;

�GHI � �JKL bySSS.

AAS

9.

10.

11.

12.

B:

IsoscelesTriangle Theorem

SAS

x � 3

m�1, m�3, m�4,m�6, and m�9each equal 20,

m�2 � 40,m�5 � 40,

m�8 � 60, andm�7 � 140

x

y

B(a � b, 0)

C(a � b, c)2

A(0, 0)


Chapter 4 Assessment Answer KeyPage 237, Open-Ended Assessment

Scoring Rubric

Score General Description Specific Criteria

• Shows thorough understanding of the concepts of usingthe Distance Formula to classify triangles and verifycongruence, finding missing angles, solving algebraicequations in isosceles and equilateral triangles, provingtriangles congruent, verifying congruence transformations,and writing coordinate proofs.

• Uses appropriate strategies to solve problems• Written explanations are exemplary.• Figures are accurate and appropriate.• Goes beyond requirements of some or all problems.

• Shows understanding of the concepts of using the DistanceFormula to classify triangles and verify congruence, findingmissing angles, solving algebraic equations in isosceles andequilateral triangles, proving triangles congruent, verifyingcongruence transformations, and writing coordinate proofs.

• Uses appropriate strategies to solve problems• Computations are mostly correct.• Written explanations are effective.• Figures are mostly accurate and appropriate.• Satisfies all requirements of all problems.

• Shows understanding of most of the concepts of using theDistance Formula to classify triangles and verifycongruence, finding missing angles, solving algebraicequations in isosceles and equilateral triangles, provingtriangles congruent, verifying congruence transformations,and writing coordinate proofs.

• 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 conceptsof using the Distance Formula to classify triangles andverify congruence, finding missing angles, solvingalgebraic equations in isosceles and equilateral triangles,proving triangles congruent, verifying congruencetransformations, and writing coordinate proofs.

• Does not use appropriate strategies to solve problems• Computations are incorrect.• Written explanations are unsatisfactory.• Figures are inaccurate or inappropriate.• Does not satisfy the requirements of the 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

Chapter 4 Assessment Answer KeyPage 237, Open-Ended Assessment

Sample Answers


1. a. The figure is an acute isosceles triangle.

b. 9x � 4 � 2(20x � 10) � 1809x � 4 � 40x � 20 � 180

49x � 16 � 18049x � 196

x � 4

2. a. To determine whether a triangle is equilateral, the coordinates shouldbe graphed first to see if they form a triangle. Then list the segmentswhich form the triangle and use the Distance Formula to find theirlengths. A triangle is equilateral only if all three sides have equalmeasures.

b. This triangle is not equilateral since AB � AC � 5 and BC � 5�2�,which makes this an isosceles triangle.

3. Since �DBA, �ABC, and �EBC form a straight line, the sum of the angle measures is 180°. Therefore m�ABC � 180 � 40 � 62 or 78.Then since the sum of the measures of the angles of a triangle is 180°,�1 � 180 � 78 � 58 or 44. Lastly, since �2 is an exterior angle, itsmeasure is equivalent to the sum of the measures of the two remoteinterior angles, 58 � 78, which equals 136.

4. a. SSS postulate

b. �J � �G, �D � �E, �L � �SD�J� � E�G�, D�L� � E�S�, J�L� � G�S�

5. Statements Reasons

1. A�B� || D�E� 1. Given

2. �ABC � �DEC 2. Alt. int. � are �.

3. A�D� bisects B�E�. 3. Given

4. B�C� � E�C� 4. Definition of segment bisector

5. �ACB � �DCE 5. Vert. � are �.

6. �ABC � �DEC 6. ASA

In addition to the scoring rubric found on page A28, the following sample answers may be used as guidance in evaluating open-ended assessment items.

An

swer

s


Chapter 4 Assessment Answer KeyVocabulary Test/Review Quiz 1 Quiz 3Page 238 Page 239 Page 240

Quiz 2Page 239

Quiz 4Page 240

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

equiangulartriangle

obtuse triangle

remote interiorangles

base

scalene triangle

flow proof

congruencetransformation

included side

coordinate proof

vertex angle

a statement that caneasily be provedusing a theorem

two triangles in whichall corresponding

parts are congruent

a triangle in which allangle measures arebetween 0 and 90

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

right scalene

C

x � 2, AB � 15,BC � 15, AC � 10

AB � �53�, BC �

2�10�, AC � �41�;scalene

4045

135737030

1.

2.

3.

4.

�KMN � �KML

D

AB � A�B� � �13�,

BC � B�C� � 2�5�,AC � A�C� � �17�

SAS

1.

2.

3.

4.

Def. of anglebisector

AAS

60

45

1.

2.

3.

4.

5.

I(0, c) and C(b, 0)

AC � �34�, AB � 6,

and CB � �34�

A�C� � C�B�

x

y

S(1–2b, 0)

G(1–4b, c)

E(0, 0)

x

y

J(a–2, 0)

D(0, a)

L(0, 0)


Chapter 4 Assessment Answer KeyMid-Chapter Test Cumulative ReviewPage 241 Page 242

An

swer

s

Part I

Part II

5.

6.

7.

AB � �41�,

BC � �29�,AC � 5�2�; scalene

AB � A�B� � �26�,

BC � B�C � � 2�5�,AC � A�C � � 3�2�

P�O� and L�N� bisecteach other.

1.

2.

3.

4.

D

C

A

B

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

15.

a ray

35�12

� in. to 36 �12

� in.

5

�4

�17

Sometimes; D, E,and F can benoncollinear.

always

undefined

4

F�D�

right triangle

15

�P � �H, �Q � �G,

�R � �B, P�Q� � H�G�,Q�R� � G�B�, P�R� � H�B�

ASA

E(b, b); F(2b, 0);G(b, 0)


Chapter 4 Assessment Answer KeyStandardized Test Practice

Page 243 Page 244

1.

2.

3.

4.

5.

6.

7. 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 D8. 9.

10. 11.

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

12.

13.

14.

15.

29 ft

35

62

40

0 6 8

6 1 8

Documents

Chapter 4 Resource Masters - Math Problem Solvingjaeproblemsolving.weebly.com/.../5/1/...chapter_4.pdf · ©Glencoe/McGraw-Hill iv Glencoe Geometry Teacher’s Guide to Using the