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Chapter 4 Resource Masters

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

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

ISBN: 978-0-07-890513-1MHID: 0-07-890513-3

Printed in the United States of America.

1 2 3 4 5 6 7 8 9 10 024 14 13 12 11 10 09 08

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

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

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

Answers for Workbooks The answers for Chapter 4 of these workbooks can be found in the back of this Chapter Resource Masters booklet.

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

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

Spanish Assessment Masters (ISBN10: 0-07-890856-6, ISBN13: 978-0-07-890856-9) These masters contain a Spanish version of Chapter 4 Test Form 2A and Form 2C.

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iii

ContentsTeacher’s Guide to Using the Chapter 4

Resource Masters .........................................iv

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

Lesson 4-1Classifying TrianglesStudy Guide and Intervention ............................ 5Skills Practice .................................................... 7Practice .............................................................. 8Word Problem Practice ..................................... 9Enrichment ...................................................... 10

Lesson 4-2Angles of TrianglesStudy Guide and Intervention .......................... 11Skills Practice .................................................. 13Practice ............................................................ 14Word Problem Practice ................................... 15Enrichment ...................................................... 16Cabri Junior. ................................................... 17Geometer’s Sketchpad Activity ....................... 18

Lesson 4-3Congruent TrianglesStudy Guide and Intervention .......................... 19Skills Practice .................................................. 21Practice ............................................................ 22Word Problem Practice ................................... 23Enrichment ...................................................... 24

Lesson 4-4Proving Triangles Congruent—SSS, SASStudy Guide and Intervention .......................... 25Skills Practice .................................................. 27Practice ............................................................ 28Word Problem Practice ................................... 29Enrichment ...................................................... 30

Lesson 4-5Proving Triangles Congruent—ASA, AASStudy Guide and Intervention .......................... 31Skills Practice .................................................. 33Practice ............................................................ 34Word Problem Practice ................................... 35Enrichment ...................................................... 36

Lesson 4-6Isosceles and Equilateral TrianglesStudy Guide and Intervention .......................... 37Skills Practice .................................................. 39Practice ............................................................ 40Word Problem Practice ................................... 41Enrichment ...................................................... 42

Lesson 4-7Congruence TransformationsStudy Guide and Intervention .......................... 43Skills Practice .................................................. 45Practice ............................................................ 46Word Problem Practice ................................... 47Enrichment ...................................................... 48

Lesson 4-8Triangles and Coordinate ProofStudy Guide and Intervention .......................... 49Skills Practice .................................................. 51Practice ............................................................ 52Word Problem Practice ................................... 53Enrichment ...................................................... 54

AssessmentStudent Recording Sheet ................................ 55Rubric for Extended Response ....................... 56Chapter 4 Quizzes 1 and 2 ............................. 57Chapter 4 Quizzes 3 and 4 ............................. 58Chapter 4 Mid-Chapter Test ............................ 59Chapter 4 Vocabulary Test ............................. 60Chapter 4 Test, Form 1 ................................... 61Chapter 4 Test, Form 2A ................................. 63Chapter 4 Test, Form 2B ................................. 65Chapter 4 Test, Form 2C ................................ 67Chapter 4 Test, Form 2D ................................ 69Chapter 4 Test, Form 3 ................................... 71Chapter 4 Extended-Response Test ............... 73Standardized Test Practice ............................. 74

Answers ........................................... A1–A40

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Teacher’s Guide to Using theChapter 4 Resource Masters

The Chapter 4 Resource Masters includes the core materials needed for 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 on the TeacherWorks PlusTM CD-ROM.

Chapter ResourcesStudent-Built Glossary (pages 1–2) These masters are a student study tool that presents up to twenty of the key vocabulary terms from the chapter. Students are to record definitions and/or examples for each term. You may suggest that students highlight or star the terms with which they are not familiar. Give this to students before beginning Lesson 4-1. Encourage them to add these pages to their mathematics study notebooks. Remind them to complete the appropriate words as they study each lesson.

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

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

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

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

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

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

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

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Assessment OptionsThe assessment masters in the Chapter 4 Resource Masters offer a wide range of assessment tools for formative (monitoring) assessment and summative (final) assessment.

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

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

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

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

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

Leveled Chapter Tests• Form 1 contains multiple-choice

questions and is intended for use with below grade level students.

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

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

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

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

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

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

Answers• The answers for the Anticipation Guide

and Lesson Resources are provided as reduced pages.

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

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Chapter 4 1 Glencoe Geometry

Student-Built Glossary

This 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. Remember to add the page number where you found the term. Add these pages to your Geometry Study Notebook to review vocabulary at the end of the chapter.

4

Vocabulary TermFound

on PageDefinition/Description/Example

auxiliary line

base angle

congruent

corresponding parts

congruent polygons

equiangular triangle

equilateral triangle

image

included angle

(continued on the next page)

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Student-Built Glossary (continued)

Vocabulary TermFound

on PageDefinition/Description/Example

included side

isosceles triangle

preimage

reflection

rotation

scalene (SKAY·leen) triangle

transformation

translation

4


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Anticipation GuideCongruent Triangles

Before you begin Chapter 4

• Read each statement.

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

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

After you complete Chapter 4

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

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

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

4

Step 1

STEP 1A, D, or NS

StatementSTEP 2A or D

1. For a triangle to be acute, all three angles must be acute.

2. All sides of an equilateral triangle are congruent.

3. An exterior angle is any angle outside of the given triangle.

4. The sum of the measures of the angles in a triangle is 360°.

5. An obtuse triangle is a triangle with three obtuse angles.

6. Triangles that are the same shape and size are congruent.

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

8. An isosceles triangle is a triangle with two congruent sides.

9. A triangle can be equiangular and not equilateral.

10. To make the calculations in a coordinate proof easier, place either a vertex or the center of the polygon at the origin.

Step 2

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Ejercicios preparatoriosTriá ngulos congruentes

Antes de comenzar el Capítulo 4

• Lee cada enunciado.

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

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

Después de completar el Capítulo 4

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

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

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

Paso 2

Paso 1

PASO 1A, D, o NS

EnunciadoPASO 2A o D

1. Para que un triángulo sea acutángulo, todos sus tres ángulos deben ser agudos.

2. Todos los lados de un triángulo equilátero son iguales.

3. Un ángulo exterior es cualquier ángulo fuera del triángulo dado.

4. La suma de las medidas de los ángulos de un triángulo es 360°.

5. Un triángulo obtusángulo es un triángulo con tres ángulos obtusos.

6. Los triángulos que son iguales en forma y tamaño son congruentes.

7. Si los ángulos de dos triángulos son congruentes, entonces los dos triángulos son congruentes.

8. Un triángulo isósceles es un triángulo con dos lados congruentes.

9. Un triángulo puede ser equiángulo y no equilátero.

10. Para facilitar los cálculos en una demostración por coordenadas, ubica ya sea un vértice o el centro del polígono en el origen.

4

Capítulo 4 4 Geometría de Glencoe


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Study Guide and InterventionClassifying Triangles

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

• 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.

• 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.

Classify each triangle.

a.

60°

A

B C

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

b.

25°35°

120°

D F

E

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

c. 90°

60° 30°

G

H J

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

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

1. 67°

90° 23°

K

L M

2.

120°

30° 30°

N O

P

3.

60° 60°

60°

Q

R S

4.

65° 65°

50°

U V

T 5. 45°

45°90°

X Y

W 6. 60°

28° 92°F D

B

4-1

Example


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Study Guide and Intervention (continued)

Classifying Triangles

Classify Triangles by Sides You can classify a triangle by the number of congruent 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.

Equilateral triangles can also be considered isosceles.

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

Classify each triangle.

a.

L J

H b. N

R P

c. 23

12

15X V

T

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.

ExercisesClassify each triangle as equilateral, isosceles, or scalene.

1. 2

1

3√⎯G C

A 2. G

K I18

18 18

3. 12 17

19Q O

M

4.

UW

S 5. 8x

32x

32x

B

CA

6. D E

F

x

x x

Example

4-1

7. ALGEBRA Find x and the length of each side if �RST is an equilateral triangle.

8. ALGEBRA Find x and the length of each side if �ABC is isosceles with AB = BC.

5x - 42x + 2

3x

3y + 24y

3y


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Skills PracticeClassifying Triangles

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

1.

60° 60°

60°

2. 40°

95° 45°

3.

50° 40°

90°

4. 80°

50°

50°

5.

100°

30°

50°

6. 70°

55° 55°

Classify each triangle as equilateral, isosceles, or scalene.

E CD

A B9

8

8√2 7. �ABE 8. �EDB

9. �EBC 10. �DBC

11. ALGEBRA Find x and the length of each side if �ABC is an isosceles triangle with

−− AB �

−− BC .

12. ALGEBRA Find x and the length of each side if �FGH is an equilateral triangle.

5x - 83x + 10

4x + 1

13. ALGEBRA Find x and the length of each side if �RST is an isosceles triangle with

−− RS �

−− TS .

3x - 1

9x + 2

6x + 8

14. ALGEBRA Find x and the length of each side if �DEF is an equilateral triangle.

5x - 212x + 6

7x - 39

4-1

4x - 3 2x + 5

x + 2


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PracticeClassifying Triangles

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

1. 2. 3.

Classify each triangle in the figure at the right by its angles and sides.

4. �ABD 5. �ABC

6. �EDC 7. �BDC

ALGEBRA For each triangle, find x and the measure of each side.

8. �FGH is an equilateral triangle with FG = x + 5, GH = 3x - 9, and FH = 2x - 2.

9. �LMN is an isosceles triangle, with LM = LN, 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

60°30°

90°

65°30°

85°

40°

100°

40°

4-1


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Word Problem PracticeClassifying Triangles

1. MUSEUMS Paul is standing in front of a museum exhibition. When he turns his head 60° to the left, he can see a statue by Donatello. When he turns his head 60° to the right, he can see a statue by Della Robbia. The two statues and Paul form the vertices of a triangle. Classify this triangle as acute, right, or obtuse.

2. PAPER Marsha cuts a rectangular piece of paper in half along a diagonal. The result is two triangles. Classify these triangles as acute, right, or obtuse.

3. WATERSKIING Kim and Cassandra are waterskiing. They are holding on to ropes that are the same length and tied to the same point on the back of a speed boat. The boat is going full speed ahead and the ropes are fully taut.

Kim, Cassandra, and the point where the ropes are tied on the boat form the vertices of a triangle. The distance between Kim and Cassandra is never equal to the length of the ropes. Classify the triangle as equilateral, isosceles, or scalene.

4. BOOKENDS Two bookends are shaped like right triangles.

The bottom side of each triangle is exactly half as long as the slanted side of the triangle. If all the books between the bookends are removed and they are pushed together, they will form a single triangle. Classify the triangle that can be formed as equilateral, isosceles, or scalene.

5. DESIGNS Suzanne saw this pattern on a pentagonal floor tile. She noticed many different kinds of triangles were created by the lines on the tile.

a. Identify five triangles that appear to be acute isosceles triangles.

b. Identify five triangles that appear to be obtuse isosceles triangles.

4-1

Kim

Cassandra

CAG H

IF

J

B

E D


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Enrichment

Reading MathematicsWhen you read geometry, you may need to draw a diagram to make the text easier 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 is greater than the x-coordinate of A. Both coordinates of C are greater than the corresponding coordinates of B. Is �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 ABC must be obtuse.

ExercisesDraw a simple triangle on grid paper 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 �JKL acute,and S. The x-coordinate of T is less right, or obtuse? than the x-coordinate of R. Is angle R of �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, and opposites. 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 �GHI scalene, isosceles, or

equilateral?

BA

C

x

y

O

4-1

Example


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Study Guide and InterventionAngles of Triangles

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

Triangle Angle Sum

Theorem

The sum of the measures of the angles of a triangle is 180.

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 Triangle Angle -

Sum Theorem

25 + 35 + m∠T = 180 Substitution

60 + m∠T = 180 Simplify.

m∠T = 120 Subtract 60 from

each side.

Find the missing angle measures.

m∠1 + m∠A + m∠B = 180 Triangle Angle - Sum

Theorem

m∠1 + 58 + 90 = 180 Substitution

m∠1 + 148 = 180 Simplify.

m∠1 = 32 Subtract 148 from

each side.

m∠2 = 32 Vertical angles are

congruent.

m∠3 + m∠2 + m∠E = 180 Triangle Angle - Sum

Theorem

m∠3 + 32 + 108 = 180 Substitution

m∠3 + 140 = 180 Simplify.

m∠3 = 40 Subtract 140 from

each side.

58°

90°

108°

12 3

E

DAC

B

ExercisesFind the measures of each numbered angle.

1. 2.

3. 4.

5. 6. 20°

152°

DG

A

130°60°

1 2

S

R

TW

V

W T

U

30°

60°

2

1

S

Q R30°

1

90°

62°

1 N

M

P

4-2

Example 1

35°

25°R T

S

Example 2

Q

O

NM

P58°

66°

50°

321


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Angles of Triangles

Exterior Angle Theorem At each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. In the diagram below, ∠B and ∠A are the remote interior angles for exterior ∠DCB.

Exterior Angle

Theorem

The measure of an exterior angle of a triangle is equal to

the sum of the measures of the two remote interior angles.

AC

B

D1m∠1 = m∠A + m∠B

Find m∠1.

m∠1 = m∠R + m∠S Exterior Angle Theorem

= 60 + 80 Substitution

= 140 Simplify.

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°

Exercises

Find the measures of each numbered angle.

1. 2.

3. 4.

Find each measure.

5. m∠ABC 6. m∠FE

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

4-2

Example 1 Example 2


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Skills PracticeAngles of Triangles

Find the measure of each numbered angle.

1. 2.

Find each measure.

3. m∠1

4. m∠2

5. m∠3

Find each measure.

6. m∠1

7. m∠2

8. m∠3

Find each measure.

9. m∠1

10. m∠2

11. m∠3

12. m∠4

13. m∠5

Find each measure.

14. m∠1

15. m∠2 63°

1

2D

A C

B

80°

60°

40°

105°1 4 5

2

3

150°55°

70°

1 2

3

85° 55°

40°

1 2

3

146°1 2

TIGERS80°

73°

1

4-2


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PracticeAngles of Triangles

Find the measure of each numbered angle.

1. 2.

Find each measure.

3. m∠1

4. m∠2

5. m∠3

Find each measure.

6. m∠1

7. m∠4

8. m∠3

9. m∠2

10. m∠5

11. m∠6

Find each measure.

12. m∠1

13. m∠2

14. CONSTRUCTION The diagram shows an example of the Pratt Truss used in bridge construction. Use the diagram to find m∠1.

145°

1

64°

58°

1

2A

BC

D

118°

36°

68°

70°

65°

82°

1

2

3 4

5

6

58°

39°

35°

12

3

40°

1

55°

72°

1

2

4-2


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Word Problem PracticeAngles of Triangles

62˚136˚

1. PATHS Eric walks around a triangular path. At each corner, he records the measure of the angle he creates.

He makes one complete circuit around the path. What is the sum of the three angle measures that he wrote down?

2. STANDING Sam, Kendra, and Tony are standing in such a way that if lines were drawn to connect the friends they would form a triangle.

If Sam is looking at Kendra he needs to turn his head 40° to look at Tony. If Tony is looking at Sam he needs to turn his head 50° to look at Kendra. How many degrees would Kendra have to turn her head to look at Tony if she is looking at Sam?

3. TOWERS A lookout tower sits on a network of struts and posts. Leslie measured two angles on the tower.

What is the measure of angle 1?

4. ZOOS The zoo lights up the chimpanzee pen with an overhead light at night. The cross section of the light beam makes an isosceles triangle.

The top angle of the triangle is 52 and the exterior angle is 116. What is the measure of angle 1?

5. DRAFTING Chloe bought a drafting table and set it up so that she can draw comfortably from her stool. Chloe measured the two angles created by the legs and the tabletop in case she had to dismantle the table.

a. Which of the four numbered angles can Chloe determine by knowing the two angles formed with the tabletop? What are their measures?

b. What conclusion can Chloe make about the unknown angles before she measures them to find their exact measurements?

4-2

52˚

116˚1

Kendra

Tony

Sam

29˚68˚

12

3 4


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Enrichment

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

In triangle ABC, m∠A is twice m∠B, and m∠C is 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 = 180 2x + x + (x + 8) = 180 4x + 8 = 180 4x = 172

x = 43So, 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 twice m∠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.

4-2

Example


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Graphing Calculator ActivityCabri Junior: Angles of Triangles

Cabri Junior can be used to investigate the relationships between the angles in a triangle.

Use Cabri Junior to investigate the sum of the measures of the angles of a triangle and to investigate the measure of the exterior angle in a triangle in relation to its remote interior angles.

Step 1 Use Cabri Junior to draw a triangle. • Select F2 Triangle to draw a triangle. • Move the cursor to where you want the first vertex. Then press ENTER . • Repeat this procedure to determine the next two vertices of the triangle.

Step 2 Label each vertex of the triangle. • Select F5 Alph-num. • Move the cursor to a vertex and press ENTER . Enter A and press ENTER again. • Repeat this procedure to label vertex B and vertex C.

Step 3 Draw an exterior angle of �ABC. • Select F2 Line to draw a line through

−−− BC .

• Select F2 Point, Point on to draw a point on � �� BC so that C is between B and the new point.

• Select F5 Alph-num to label the point D.

Step 4 Find the measure of the three interior angles of �ABC and the exterior angle, ∠ACD.

• Select F5 Measure, Angle. • To find the measure of ∠ABC, select points A, B, and C

(with the vertex B as the second point selected). • Use this method to find the remaining angle measures.

Exercises

Analyze your drawing.

1. Use F5 Calculate to find the sum of the measures of the three interior angles.

2. Make a conjecture about the sum of the interior angles of a triangle.

3. Change the dimensions of the triangle by moving point A. (Press CLEAR so the pointer becomes a black arrow. Move the pointer close to point A until the arrow becomes transparent and point A is blinking. Press ALPHA to change the arrow to a hand. Then move the point.) Is your conjecture still true?

4. What is the measure of ∠ACD? What is the sum of the measures of the two remote interior angles (∠ABC and ∠BAC)?

5. Make a conjecture about the relationship between the measure of an exterior angle of a triangle and its two remote interior angles.

6. Change the dimensions of the triangle by moving point A. Is your conjecture still true?

4-2

Example


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Geometer’s Sketchpad ActivityAngles of Triangles

The Geometer’s Sketchpad can be used to investigate the relationships between the angles in a triangle.

Use The Geometer’s Sketchpad to investigate the sum of the measures of the angles of a triangle and to investigate the measure of the exterior angle in a triangle in relation to its remote interior angles.

Step 1 Use The Geometer’s Sketchpad to draw a triangle. • Construct a ray by selecting the Ray tool from the toolbar. First, click where

you want the first point. Then click on a second point to draw the ray. • Next, select the Segment tool from the toolbar. Use the endpoint of the ray as

the first point for the segment and click on a second point to construct the segment.

• Construct another segment joining the second point of the new segment to a point on the ray.

Step 2 Display the labels for each point. Use the pointer tool to select all four points. Display the labels by selecting Show Labels from the Display menu.

Step 3 Find the measures of the three interior angles and the one exterior angle. To find the measure of angle ABC, use the Selection Arrow tool to select points A, B, and C (with the vertex B as the second point selected). Then, under the Measure menu, select Angle. Use this method to find the remaining angle measures.

ExercisesAnalyze your drawing.

1. What is the sum of the measures of the three interior angles?

2. Make a conjecture about the sum of the interior angles of a triangle.

3. Change the dimensions of the triangle by selecting point A with the pointer tool and moving it. Is your conjecture still true?

4. What is the measure of ∠ACD? What is the sum of the measures of the two remote interior angles (∠ABC and ∠BAC)?

5. Make a conjecture about the relationship between the measure of an exterior angle of a triangle and its two remote interior angles.

6. Change the dimensions of the triangle by selecting point A with the pointer tool and moving it. Is your conjecture still true?

m/ABC = 60.00°m/BAC = 58.19°m/ACB = 61.81°m/ACD = 118.19°m/ABC + m/BAC + m/ACB = 180.00°m/ABC + m/BAC = 118.19°

B C D

A

4-2

Example


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Congruence and Corresponding Parts Triangles 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 � ∠T −−

XY � −−

RS , −−

XZ � −−

RT , −−

YZ � −−

ST

ExercisesShow that the polygons are congruent by identifying all congruent corresponding parts. Then write a congruence statement.

1.

CA

B

LJ

K 2.

C

D

A

B 3. K

J

L

M

4. F G L K

JE

5. B D

CA

6. R

T

U S

7. Find the value of x.

8. Find the value of y.

Y

XZ

T

SR

AC

B

R

T

S

4-3 Study Guide and InterventionCongruent Triangles

Third Angles

Theorem

If two angles of one triangle are congruent to two angles

of a second triangle, then the third angles of the triangles are congruent.

Example

2x + y(2y - 5)°75°

65° 40°96.6

90.664.3


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Congruent Triangles

4-3

Example

Prove Triangles Congruent Two triangles are congruent if and only if their corresponding parts are congruent. Corresponding parts include corresponding angles and corresponding sides. The phrase “if and only if” means that both the conditional and its converse are true. For triangles, we say, “Corresponding parts of congruent triangles are congruent,” or CPCTC.

Write a two-column proof.

Given: −−

AB � −−−

CB , −−−

AD � −−−

CD , ∠BAD � ∠BCD −−−

BD bisects ∠ABC Prove: �ABD � �CBD

Proof:Statement Reason

1. −−

AB � −−−

CB , −−−

AD � −−−

CD 2.

−−− BD �

−−− BD

3. ∠BAD � ∠BCD4. ∠ABD � ∠CBD5. ∠BDA � ∠BDC6. �ABD � �CBD

1. Given2. Reflexive Property of congruence3. Given4. Definition of angle bisector

5. Third Angles Theorem6. CPCTC

ExercisesWrite a two-column proof. 1. Given: ∠A � ∠C, ∠D � ∠B,

−−− AD �

−−− CB ,

−− AE �

−−− CE ,

−−

AC bisects −−−

BD Prove: �AED � �CEB

Write a paragraph proof. 2. Given:

−−− BD bisects ∠ABC and ∠ADC,

−−

AB � −−−

CB , −−

AB � −−−

AD , −−−

CB � −−−

DC Prove: �ABD � �CBD


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4-3

Show that polygons are congruent by identifying all congruent corresponding parts. Then write a congruence statement.

1.

L

P

J

S

V

T

2.

In the figure, �ABC � �FDE. 3. Find the value of x.


5. PROOF Write a two-column proof.

Given: −−−

AB � −−−

CB , −−−

AD � −−−

CD , ∠ABD � ∠CBD,∠ADB � ∠CDB

Prove: �ABD � �CBD

Skills PracticeCongruent Triangles

D

E

G

F

108° 48° y°

(2x - y)°

A F

BC ED


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PracticeCongruent Triangles

Show that the polygons are congruent by indentifying all congruent corresponding parts. Then write a congruence statement.

1.

D

R

SCA

B 2. MN

L

P

Q

Polygon ABCD � polygon PQRS.



5. PROOF Write a two-column proof.Given: ∠P � ∠R, ∠PSQ � ∠RSQ,

−−− PQ �

−−− RQ ,

−−

PS � −−

RS

Prove: �PQS � �RQS

6. QUILTING

a. Indicate the triangles that appear to be congruent.

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

B

A

I

E

FH G

C D

4-3

(2x + 4)°

(3y - 3)

80°

100°

10

12


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Word Problem PracticeCongruent Triangles

1. PICTURE HANGING Candice hung a picture that was in a triangular frame on her bedroom wall.

Before

A B

C

After

One day, it fell to the floor, but did not break or bend. The figure shows the object before and after the fall. Label the vertices on the frame after the fall according to the “before” frame.

2. SIERPINSKI’S TRIANGLE The figure below is a portion of Sierpinski’s Triangle. The triangle has the property that any triangle made from any combination of edges is equilateral. How many triangles in this portion are congruent to the black triangle at the bottom corner?

3. QUILTING Stefan drew this pattern for a piece of his quilt. It is made up of congruent isosceles right triangles. He drew one triangle and then repeatedly drew it all the way around.

45˚

What are the missing measures of the angles of the triangle?

4. MODELS Dana bought a model airplane kit. When he opened the box, these two congruent triangular pieces of wood fell out of it.

A

B

CQ

R

P

Identify the triangle that is congruent to �ABC.

5. GEOGRAPHY Igor noticed on a map that the triangle whose vertices are the supermarket, the library, and the post office (�SLP) is congruent to the triangle whose vertices are Igor’s home, Jacob’s home, and Ben’s home (�IJB). That is, �SLP � �IJB.

a. The distance between the supermarket and the post office is 1 mile. Which path along the triangle �IJB is congruent to this?

b. The measure of ∠LPS is 40. Identify the angle that is congruent to this angle in �IJB.

4-3


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Enrichment4-3

Overlapping TrianglesSometimes when triangles overlap each other, they share corresponding parts. When trying to picture the corresponding parts, try redrawing the triangles separately.

In the figure �ABD � �DCA and m∠BDA = 30°.Find m∠CAD and m∠CDA.

30°

First, redraw �ABD and �DCA like this.

30°

Now you can see that ∠BDA � ∠CAD because they are corresponding parts of congruent triangles, so m∠CAD = 30. Using the Triangle Sum Theorem, 180 - (90 + 30) = 60, so m∠CDA = 60.

Exercises 1. In the figure �ABC � �CDB, m∠ABC = 110, and

m∠D = 20. Find m∠ACB and m∠DCB.

2. Write a two column proof.Given: ∠A and ∠B are right angles, ∠ACD � ∠BDC,

−−−

AD � −−−

BC , −−

AC � −−−

BD

Prove: �ADC � �CBD

Example


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Study Guide and InterventionProving Triangles Congruent—SSS, SAS

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


Given: −−

AB � −−−

DB and C is the midpoint of −−−

AD .Prove: �ABC � �DBC B

CDA

Statements Reasons

1. −−

AB � −−−

DB 1. Given2. C is the midpoint of

−−− AD . 2. Given

3. −−

AC � −−−

DC 3. Midpoint Theorem4.

−−− BC �

−−− BC 4. Reflexive Property of �

5. �ABC � �DBC 5. SSS Postulate

Exercises


1. B Y

CA XZ

Given: AB �−−XY ,

−−AC �

−−XZ ,

−−−BC �

−−YZ

Prove: �ABC � �XYZ

2. T U

R S

Given: −−

RS � −−−

UT , −−

RT � −−−

US Prove: �RST � �UTS

4-4

SSS PostulateIf three sides of one triangle are congruent to three sides of a second

triangle, then the triangles are congruent.

Example


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Proving Triangles Congruent—SSS, SAS

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

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

a. A

B C

X

Y Z

b. D H

F E

G

J

c. P Q

S

1

2 R

In �ABC, the angle is not The right angles are The included angles, “included” by the sides

−− AB congruent and they are the ∠1 and ∠2, are congruent

and −−

AC . 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.

Exercises

4-4

SAS Postulate

If 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.

Example

Write the specified type of proof. 1. Write a two column proof. 2. Write a two-column proof. Given: NP = PM,

−−− NP ⊥

−− PL Given: AB = CD,

−− AB ‖

−−− CD

Prove: �NPL � �MPL Prove: �ACD � �CAB

3. Write a paragraph proof.Given: V is the midpoint of

−− YZ .

V is the midpoint of −−−

WX .Prove: �XVZ � �WVY


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Determine whether �ABC � �KLM. 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)

PROOF Write the specified type of proof.

3. Write a flow proof. Given:

−− PR �

−−− DE ,

−− PT �

−−− DF

∠R � ∠E, ∠T � ∠F Prove: �PRT � �DEF

4. Write a two-column proof.Given:

−− AB �

−−− CB , D is the midpoint of

−− AC .

Prove: �ABD � �CBD

T

R

P

F

E

D

4-4 Skills PracticeProving Triangles Congruent—SSS, SAS


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PracticeProving Triangles Congruent—SSS, SAS

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:

−− RS �

−− TS

V is the midpoint of −−

RT .Prove: �RSV � �TSV

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove congruence, 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 the diagram. How does he know that the lengths A'B' and AB are equal? A'B'

A B

C

S

R

V

T

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Word Problem PracticeProving Triangles Congruent—SSS, SAS

1. STICKS Tyson had three sticks of lengths 24 inches, 28 inches, and 30 inches. Is it possible to make two noncongruent triangles using the same three sticks? Explain.

2. BAKERY Sonia made a sheet of baklava. She has markings on her pan so that she can cut them into large squares. After she cuts the pastry in squares, she cuts them diagonally to form two congruent triangles. Which postulate could you use to prove the two triangles congruent?

3. CAKE Carl had a piece of cake in the shape of an isosceles triangle with angles 26, 77, and 77. He wanted to divide it into two equal parts, so he cut it through the middle of the 26 angle to the midpoint of the opposite side. He says that because he is dividing it at the midpoint of a side, the two pieces are congruent. Is this enough information? Explain.

4. TILES Tammy installs bathroom tiles. Her current job requires tiles that are equilateral triangles and all the tiles have to be congruent to each other. She has a big sack of tiles all in the shape of equilateral triangles. Although she knows that all the tiles are equilateral, she is not sure they are all the same size. What must she measure on each tile to be sure they are congruent? Explain.

5. INVESTIGATION An investigator at a crime scene found a triangular piece of torn fabric. The investigator remembered that one of the suspects had a triangular hole in their coat. Perhaps it was a match. Unfortunately, to avoid tampering with evidence, the investigator did not want to touch the fabric and could not fit it to the coat directly.

a. If the investigator measures all three side lengths of the fabric and the hole, can the investigator make a conclusion about whether or not the hole could have been filled by the fabric?

b. If the investigator measures two sides of the fabric and the included angle and then measures two sides of the hole and the included angle can he determine if it is a match? Explain.

4-4


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Enrichment4-4

Congruent Triangles in the Coordinate Plane

If you know the coordinates of the vertices of two triangles in the coordinate plane, you can often decide whether the two triangles are congruent. There may 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 you know 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.


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Study Guide and InterventionProving Triangles Congruent—ASA, AAS

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

Write a two column proof.

Given: −−

AB ‖ −−−

CD ∠CBD � ∠ADB Prove: �ABD � �CDBStatements Reasons1.

−− AB ‖

−−− CD

2. ∠CBD � ∠ADB3. ∠ABD � ∠BDC4. BD = BD5. �ABD � �CDB

1. Given2. Given3. Alternate Interior Angles Theorem4. Reflexive Property of congruence5. ASA

4-5

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.

Example

ExercisesPROOF Write the specified type of proof. 1. Write a two column proof. 2. Write a paragraph proof.

S

V

U

R T

A C

B

E

D

Given: ∠S � ∠V, Given: −−−

CD bisects −−

AE , −−

AB ‖ −−−

CD T is the midpoint of

−− SV ∠E � ∠BCA

Prove: �RTS � �UTV Prove: �ABC � �CDE


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Proving Triangles Congruent—ASA, AAS

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

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 Theorem?

−−

AC � −−

AC by the Reflexive Property of congruence. The congruent angles cannot be ∠1 and ∠2, because

−− AC would be the included side.

If ∠B � ∠D, then �ABC � �ADC by the AAS Theorem.

ExercisesPROOF Write the specified type of proof. 1. Write a two column proof. Given:

−−− BC ‖

−− EF

AB = ED ∠C � ∠F Prove: �ABD � �DEF

2. Write a flow proof. Given: ∠S � ∠U;

−− TR bisects ∠STU.

Prove: ∠SRT � ∠URT

S

R T

U

4-5

AAS TheoremIf two angles and a nonincluded side of one triangle are congruent to the corresponding

two angles and side of a second triangle, then the two triangles are congruent.

Example

D

C12A

B


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PROOF Write a flow proof.

1. Given: ∠N � ∠L

−− JK �

−−− MK

Prove: �JKN � �MKL

2. Given: −−

AB � −−−

CB ∠A � ∠C

−−− DB bisects ∠ABC.

Prove: −−−

AD � −−−

CD

3. Write a paragraph proof. Given:

−−− DE ‖

−−− FG

∠E � ∠G Prove: �DFG � �FDE

FG

D E

A C

B

D

N

J

M

K L

4-5 Skills PracticeProving Triangles Congruent—ASA, AAS


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PROOF Write the specified type of proof. 1. Write a flow proof. Given: S is the midpoint of

−−−QT .

−−−

QR ‖ −−−

TU Prove: �QSR � �TSU

2. Write a paragraph proof. Given: ∠D � ∠F

−−−

GE bisects ∠DEF. Prove:

−−− DG �

−−− FG

ARCHITECTURE For Exercises 3 and 4, use the following information.An architect used the window design in the diagram when remodeling an art studio.

−− AB and

−−− CB each measure 3 feet.

3. Suppose D is the midpoint of −−

AC . 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

UQ S

RT

4-5 PracticeProving Triangles Congruent—ASA, AAS


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1. DOOR STOPS Two door stops have cross-sections that are right triangles. They both have a 20 angle and the length of the side between the 90 and 20 angles are equal. Are the cross-sections congruent? Explain.

2. MAPPING Two people decide to take a walk. One person is in Bombay and the other is in Milwaukee. They start by walking straight for 1 kilometer. Then they both turn right at an angle of 110, and continue to walk straight again. After a while, they both turn right again, but this time at an angle of 120. They each walk straight for a while in this new direction until they end up where they started. Each person walked in a triangular path at their location. Are these two triangles congruent? Explain.

3. CONSTRUCTION The rooftop of Angelo’s house creates an equilateral triangle with the attic floor. Angelo wants to divide his attic into 2 equal parts. He thinks he should divide it by placing a wall from the center of the roof to the floor at a 90 angle. If Angelo does this, then each section will share a side and have corresponding 90 angles. What else must be explained to prove that the two triangular sections are congruent?

90˚ 90˚

4. LOGIC When Carolyn finished her musical triangle class, her teacher gave each student in the class a certificate in the shape of a golden triangle. Each student received a different shaped triangle. Carolyn lost her triangle on her way home. Later she saw part of a golden triangle under a grate.

Is enough of the triangle visible to allow Carolyn to determine that the triangle is indeed hers? Explain.

5. PARK MAINTENANCE Park officials need a triangular tarp to cover a field shaped like an equilateral triangle 200 feet on a side.

a. Suppose you know that a triangular tarp has two 60 angles and one side of length 200 feet. Will this tarp cover the field? Explain.

b. Suppose you know that a triangular tarp has three 60 angles. Will this tarp necessarily cover the field? Explain.

4-5 World Problem PraticeProving Triangles Congruent—ASA, AAS


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Enrichment4-5

What About AAA to Prove Triangle Congruence?You have learned about the SSS and ASA postulates to prove that two triangles are congruent. Do you think that triangles can be proven congruent by AAA?

Investigate the AAS theorem. AAS represents the conjecture that two triangles are congruent if they have two pairs of corresponding angles congruent and corresponding non-included sides congruent. For example, in the triangles at the right, ∠A � ∠Y, ∠B � ∠X, and

−− AC �

−− YZ .

1. Refer to the drawing above. What is the measure of the missing angle in the first triangle? What is the measure of the missing angle in the second triangle?

2. With the information from Exercise 1, what postulate can you now use to prove that the two triangles are congruent?

Now investigate the AAA conjecture. AAA represents the conjecture that two triangles are congruent if they have three pairs of corresponding angles congruent. The triangles below demonstrate AAA.

25˚

25˚

70˚70˚

85˚

85˚

3. Are the triangles above congruent? Explain.

4. Can you draw two triangles that have all three angles congruent that are not congruent? Can AAA be used to prove that two triangles are congruent?

5. What characteristics do these two triangles drawn by AAA seem to possess?

89˚89˚

66˚

66˚A

B C

5 in

5 in

X

Y

Z


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ExercisesALGEBRA Find the value of each variable.

1. R

P

Q2x°

40°

2. S

V

T 3x - 6

2x + 6 3. W

Y Z3x°

4. 2x°

(6x + 6)°

5.

5x°

6.

R S

T3x°

x°


Given: ∠1 � ∠2 Prove:

−− AB �

−−− CB

Study Guide and InterventionIsosceles and Equilateral Triangles

Properties of Isosceles Triangles An isosceles triangle has two congruent sides called the legs. The angle formed by the legs is called the vertex angle. The other two angles are called base 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. (Converse of Isosceles Triangle Theorem)

If −−

AB � −−−

CB , then ∠A � ∠C.

If ∠A � ∠C, then −−

AB � −−−

CB .

A

B

C

Find x, given −−

BC � −−

BA .

B

A

C (4x + 5)°

(5x - 10)°

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.

Example 1 Example 2 Find x.

R T

S

3x - 13

2x

m∠S = m∠T, so SR = TR Converse of Isos. � Thm.

3x - 13 = 2x Substitution

3x = 2x + 13 Add 13 to each side.

x = 13 Subtract 2x from each side.

B

AC D

E

1 32

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Isosceles and Equilateral Triangles

Properties of Equilateral Triangles An equilateral triangle has three congruent sides. The Isosceles Triangle Theorem leads to two corollaries about equilateral triangles.

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; −−−

PQ ‖ −−−

BC . 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. � are �.4. m∠1 = 60, m∠2 = 60 4. Substitution5. �APQ is equilateral. 5. If a � is equiangular, then it is equilateral.

Exercises

ALGEBRA Find the value of each variable.

1. D

F E6x°

2. G

J H

6x - 5 5x

3.

3x°

4.

4x 4060°

5. X

Z Y4x - 4

3x + 8 60°

6.

4x°

60°


Given: �ABC is equilateral; ∠1 � ∠2.

Prove: ∠ADB � ∠CDB

A

D

C

B12

A

B

P Q

C

1 2

4-6

Example


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Skills PracticeIsosceles and Equilateral Triangles

D

C

B

A E

Refer to the figure at the right.

1. If −−

AC � −−−

AD , name two congruent angles.

2. If

−−− BE �

−−− BC , name two congruent angles.

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

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

Find each measure. 5. m∠ABC 6. m∠EDF

60°

40°

ALGEBRA Find the value of each variable.

7.

2x + 4 3x - 10

8.

(2x + 3)°


Given: −−−

CD � −−−

CG

−−−

DE � −−−

GF

Prove: −−−

CE � −−

CF

D

E

FG

C

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PracticeIsosceles and Equilateral Triangles

Lincoln Hawks

E

D B

C

A1

23

4

U

R

TV

S

Refer to the figure at the right.

1. If −−−

RV � −−

RT , name two congruent angles.

2. If −−

RS � −−

SV , name two congruent angles.

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

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

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.


Given: −−−

DE ‖ −−−

BC ∠1 � ∠2

Prove: −−

AB � −−

AC

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.

4-6

50°


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Word Problem PracticeIsosceles and Equilateral Triangles

1

1. TRIANGLES At an art supply store, two different triangular rulers are available. One has angles 45°, 45°, and 90°. The other has angles 30°, 60°, and 90°. Which triangle is isosceles?

2. RULERS A foldable ruler has two hinges that divide the ruler into thirds. If the ends are folded up until they touch, what kind of triangle results?

3. HEXAGONS Juanita placed one end of each of 6 black sticks at a common point and then spaced the other ends evenly around that point. She connected the free ends of the sticks with lines. The result was a regular hexagon. This construction shows that a regular hexagon can be made from six congruent triangles. Classify these triangles. Explain.

4. PATHS A marble path is constructed out of several congruent isosceles triangles. The vertex angles are all 20. What is the measure of angle 1 in the figure?

5. BRIDGES Every day, cars drive through isosceles triangles when they go over the Leonard Zakim Bridge in Boston. The ten-lane roadway forms the bases of the triangles.

a. The angle labeled A in the picture has a measure of 67. What is the measure of ∠B?

b. What is the measure of ∠C?

c. Name the two congruent sides.

A

B

C

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Enrichment

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

1. Given: BE = BF, ∠BFG � ∠BEF � ∠BED, m∠BFE = 82 and ABFG and BCDE each have opposite sides parallel and congruent.

Find m∠ABC.

A

G DF E

CB

3. Given: m∠UZY = 90, m∠ZWX = 45, �YZU � �VWX, UVXY is a square (all sides congruent, all angles right angles).

Find m∠WZY.

2. Given: AC = AD, and −−

AB ⊥ −−−

BD , m∠DAC = 44 and −−−

CE bisects ∠ACD. Find m∠DEC.

A

D C

BE

4. Given: m∠N = 120, −−−

JN � −−−

MN , �JNM � �KLM.

Find m∠JKM.

J

K

L

MN

U VW

XYZ

4-6


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Identify Congruence Transformations A congruence transformation is a transformation where the original figure, or preimage, and the transformed figure, or image, figure are still congruent. The three types of congruence transformations are reflection (or flip), translation (or slide), and rotation (or turn).

Identify the type of congruence transformation shown as a reflection, translation, or rotation.

x

y

Each vertex and its image are the

same distance from the y-axis.

This is a reflection.

x

y

Each vertex and its image are in

the same position, just two units

down. This is a translation.

x

y

Each vertex and its image are the

same distance from the origin.

The angles formed by each pair of

corresponding points and the

origin are congruent. This is a

rotation.

ExercisesIdentify the type of congruence transformation shown as a reflection, translation, or rotation.

1.

x

y 2.

x

y 3.

x

y

4.

x

y 5.

x

y 6.

x

y

Study Guide and InterventionCongruence Transformations

Example

4-7


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Verify Congruence You can verify that reflections, translations, and rotations of triangles produce congruent triangles using SSS.

Verify congruence after a transformation.

�WXY with vertices W(3, -7), X(6, -7), Y(6, -2) is a transformation of �RST with vertices R(2, 0), S(5, 0), and T(5, 5). Graph the original figure and its image. Identify the transformation and verify that it is a congruence transformation.

Graph each figure. Use the Distance Formula to show the sides are congruent and the triangles are congruent by SSS.

RS = 3, ST = 5, TR = √ �� (5 - 2) 2 + (5 - 0) 2 = √ �� 34

WX = 3, XY = 5, YW= √ �� (6 - 3) 2 + (-2- (-7)) 2 = √ �� 34

−−

RS � −−−

WX , −−

ST , � −−

XY , −−

TR � −−−

YW

By SSS, �RST � �WXY.

Exercises

COORDINATE GEOMETRY Graph each pair of triangles with the given vertices. Then identify the transformation, and verify that it is a congruence transformation.

1. A(−3, 1), B(−1, 1), C(−1, 4) 2. Q(−3, 0), R(−2, 2), S(−1, 0)

D(3, 1), E(1, 1), F(1, 4) T(2, −4), U(3, −2), V(4, −4)

x

y

x

y

4-7 Study Guide and Intervention (continued)

Congruence Transformations

Example

x

y


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4-7

Identify the type of congruence transformation shown as a reflection, translation, or rotation.

1.

x

y 2.

x

y

COORDINATE GEOMETRY Identify each transformation and verify that it is a congruence transformation.

3.

x

y 4.

x

y

COORDINATE GEOMETRY Graph each pair of triangles with the given vertices. Then, identify the transformation, and verify that it is a congruence transformation.

5. A(1, 3), B(1, 1), C(4, 1); 6. J(−3, 0), K(−2, 4), L(−1, 0); D(3, -1), E(1, -1), F(1, -4) Q(2, −4), R(3, 0), S(4, −4)

x

y

x

y

4-7 Skills PracticeCongruence Transformations


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Identify the type of congruence transformation shown as a reflection, translation, or rotation. 1.

x

y 2.

x

y

3. Identify the type of congruence transformation shown as a reflection, translation, or rotation, and verify that it is a congruence transformation.

4. ΔABC has vertices A(−4, 2), B(−2, 0), C(−4, -2). ΔDEF has vertices D(4, 2), E(2, 0), F(4, −2). Graph the original figure and its image. Then identify the transformation and verify that it is a congruence transformation.

5. STENCILS Carly is planning on stenciling a pattern of flowers along the ceiling in her bedroom. She wants all of the flowers to look exactly the same. What type of congruence transformation should she use? Why?

4-7 PracticeCongruence Transformations

x

y

x

y


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4-7 Word Problem PracticeCongruence Transformations

1. QUILTING You and your two friends visit a craft fair and notice a quilt, part of whose pattern is shown below. Your first friend says the pattern is repeated by translation. Your second friend says the pattern is repeated by rotation. Who is correct?

2. RECYCLING The international symbol for recycling is shown below. What type of congruence transformation does this illustrate?

3. ANATOMY Carl notices that when he holds his hands palm up in front of him, they look the same. What type of congruence transformation does this illustrate?

4. STAMPS A sheet of postage stamps contains 20 stamps in 4 rows of 5 identical stamps each. What type of congruence transformation does this illustrate?

5. MOSAICS José cut rectangles out of tissue paper to create a pattern. He cut out four congruent blue rectangles and then cut out four congruent red rectangles that were slightly smaller to create the pattern shown below.

a. Which congruence transformation does this pattern illustrate?

b. If the whole square were rotated 90, would the transformation still be the same?


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4-7 Enrichment

Transformations in The Coordinate PlaneThe following statement tells one way to relate points to corresponding translated 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 or moved 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 your answer.

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

2. (x, y) → (x - 4, y) 3. (x, y) → (x + 5, y + 4)

4. (x, y) → (-x , -y) 5. (x, y) → (- 1 − 2 x, y)

x

y

Ox

y

O

x

y

Ox

y

O

x

y

B

A

C

X

Z

YO

(x, y) � (x + 6, y - 3)


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Study Guide and InterventionTriangles and Coordinate Proof

Position and Label Triangles A coordinate proof uses points, distances, and slopes to prove geometric properties. The first step in writing a coordinate proof is to place a figure on the 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 computations as simple as possible.

Position an isosceles 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 a −

2 . Use b for the y-coordinate,

so the vertex is S ( a − 2 , b) .

ExercisesName the missing coordinates of each triangle.

1.

x

y

B(2p, 0)

C(?, q)

A(0, 0)

2.

x

y

S(2a, 0)

T(?, ?)

R(0, 0)

3.

x

y

G(2g, 0)

F(?, b)

E(?, ?)

Position and label each triangle on the coordinate plane.

4. isosceles triangle 5. isosceles right �DEF 6. equilateral triangle �RST with base

−− RS with legs e units long �EQI with vertex

4a units long Q(0, √ � 3 b) and sides 2b units long

x

y

x

y

x

y

x

y

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

S(a–2, b)Example

4-8


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Write Coordinate Proofs Coordinate proofs can be used to prove theorems and to verify properties. Many coordinate proofs use the Distance Formula, Slope Formula, or Midpoint 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

−− RT .

Given: Isosceles �RST; U is the midpoint of base −−

RT .Prove:

−−− SU ⊥

−− RT

Proof:U is the midpoint of

−− RT so the coordinates of U are ( -a + a −

2 , 0 + 0 −

2 ) = (0, 0). Thus

−−− SU lies on

the y-axis, and �RST was placed so −−

RT lies on the x-axis. The axes are perpendicular, so −−−

SU ⊥ −−

RT .

Exercise

PROOF Write a coordinate proof for the statement.

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

x

y

T(a, 0)U(0, 0)R(–a, 0)

S(0, c)

4-8 Study Guide and Intervention (continued)

Triangles and Coordinate Proof

Example


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1. right �FGH with legs 2. isosceles �KLP with 3. isosceles �AND witha units and b units long base

−− KP 6b units long base

−−− AD 5a units long

x

y

x

y

x

y

Name the missing coordinates of each triangle. 4.

x

y

B(2a, 0)

A(0, ?)

C(0, 0)

5.

x

y

Y(2b, 0)

Z(?, ?)

X(0, 0)

6.

x

y

N(3b, 0)

M(?, ?)

O(0, 0)

7.

x

y

Q(?, ?)

R(2a, b)

P(0, 0)

8.

x

y

P(7b, 0)

R(?, ?)

N(0, 0)

9.

x

y

U(a, 0)

T(?, ?)

S(–a, 0)

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

Given: isosceles right �ABC with ∠ABC the right angle and M the midpoint of −−

AC

Prove: −−−

BM ⊥ −−

AC

4-8 Skills PracticeTriangles and Coordinate Proof


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1. equilateral �SWY with 2. isosceles �BLP with 3. isosceles right �DGJ sides 1 −

4 a units long base

−− BL 3b units long with hypotenuse

−− DJ and

legs 2a units long

x

y

x

y

x

y

Name the missing coordinates of each triangle.

4.

x

y S(?, ?)

J(0, 0) R( b, 0)1–3

5.

x

y

C(?, 0)

E(0, ?)

B(–3a, 0)

6.

x

y

P(2b, 0)

M(0, ?)

N(?, 0)

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 part time at the mall in a music store. The mall is 2 miles west and 3 miles north of the school.

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

Given: �SKM Prove: �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)

4-8 PracticeTriangles and Coordinate Proof


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1. SHELVES Martha has a shelf bracket shaped like a right isosceles triangle. She wants to know the length of the hypotenuse relative to the sides. She does not have a ruler, but remembers the Distance Formula. She places the bracket on a coordinate grid with the right angle at the origin. The length of each leg is a. What are the coordinates of the vertices making the two acute angles?

2. FLAGS A flag is shaped like an isosceles triangle. A designer would like to make a drawing of the flag on a coordinate plane. She positions it so that the base of the triangle is on the y-axis with one endpoint located at (0, 0). She locatesthe tip of the flag at (a, b−

2 ).

What are the coordinates of the third vertex?

3. BILLIARDS The figure shows a situation on a billiard table.

(–b, ?) y

xO

(0, 0)

What are the coordinates of the cue ball before it is hit and the point where the cue ball hits the edge of the table?

4. TENTS The entrance to Matt’s tent makes an isosceles triangle. If placed on a coordinate grid with the base on the x–axis and the left corner at the origin, the right corner would be at (6, 0) and the vertex angle would be at (3, 4). Prove that it is an isosceles triangle.

5. DRAFTING An engineer is designing a roadway. Three roads intersect to form a triangle. The engineer marks two points of the triangle at (-5, 0) and (5, 0) on a coordinate plane.

a. Describe the set of points in the coordinate plane that could not be used as the third vertex of the triangle.

b. Describe the set of points in the coordinate plane that would make the vertex of an isosceles triangle together with the two congruent sides.

c. Describe the set of points in the coordinate plane that would make a right triangle with the other two points if the right angle is located at (-5, 0).

Word Problem PracticeTriangles and Coordinate Proof

4-8


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Rectangle ParadoxUse grid paper to draw the two rectangles below. Cut them into pieces along the dashed lines. Then rearrange the pieces to form a new rectangle.

1. Make a sketch of the new rectangle. Label the whole number lengths

2. What is the combined area of the two original rectangles?

3. What is the area of the new rectangle?

4. Make a conjecture as to why there is a discrepancy between the answers to Exercises 2 and 3.

5. Use a straight edge to precisely draw the four shapes that make up the larger rectangle. What does you drawing show?

6. How could you use slope to show that you drawing is accurate?

3

5

3

35

5

4-8 Enrichment


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SCORE 4 Student Recording SheetUse this recording sheet with pages 316–317 of the Student Edition.

Read each question. Then fill in the correct answer.

1. A B C D

2. F G H J

3. A B C D

4. F G H J

5. A B C D

6. F G H J

7. A B C D

Multiple Choice

Record your answer in the blank.

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

8. ———————— (grid in)

9. ———————— (grid in)

10. ———————— (grid in)

11. ————————

12. ————————

13. ———————— (grid in)

14. ————————

Extended Response

Short Response/Gridded Response

Record your answers for Question 15 on the back of this paper.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

8.

10.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9.

13.

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

. . . . .

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0


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General Scoring Guidelines

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

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

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

Exercise 15 Rubric

Score Specific Criteria

4The points for the verties of �ABC are plotted properly and the distance formula is appropritely used to find AB = BC = √ �� 4a2+b2 . Students draw the appropriate conclusion that �ABC is isosceles.

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

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

1 A correct solution with no evidence or explanation.

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

4 Rubric for Scoring Extended Response


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SCORE

SCORE

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

2. MULTIPLE CHOICE What is the best classification of this triangle by its angles and sides?

A acute isosceles C right isosceles B 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 the value of x and the length of each side of the triangle.

4. Find the measures of the sides of �ABC with vertices A(1, 5), B(3, -2), and C(-3, 0). Classify the triangle by 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

1. Identify the congruent triangles in the figure. K

MLN

2. If �JGO � �RWI, which angle corresponds to ∠I?

3. In the diagram, �ABC � �DEF. Find the values of x and y.

4. In the figure −−−

QR � −−

SR and −−−

PQ � −−

TS . Point R is the midpoint of

−− PT . Determine

which theorem or postulate can be used to show that �QRP � �SRT.

5. What is the relationship between ∠Q and ∠S?

1.

2.

3.

4.

5.

4

4

Chapter 4 Quiz 1(Lessons 4-1 and 4-2)

70°

65°

1

2 34

56 107°

43°


Q

P TR

S

3x - y

(6y - 4)°

80°

70° 30°

19.7

18.810

1.

2.

3.

4.

5. 6. 7. 8. 9. 10.

SCORE


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SCORE

A

K

Z C

1.

2.

3.

4.

5.

For Questions 1 and 2, complete the two-column proof by supplying the missing information for each corresponding location.

Given: ∠Z � ∠C −−−

AK bisects ∠ZKC.Prove: �AKZ � �AKCProof:

Statements Reasons1. ∠Z � ∠C −−−

AK bisects ∠ZKC. 1. Given2. ∠ZKA � ∠CKA 2. (Question 1)3.

−−− AK �

−−− AK 3. Reflexive Property

4. �AKZ � �AKC 4. (Question 2)

For Questions 3 and 4 refer to the figure.

3. Find m∠1 4. Find m∠2 5. The base angle of an isosceles triangle is 30.

What is the measure of the vertex angle?

1. Identify the type of congruence transformation shown as a reflection, translation, or rotation.

Position and label the following triangle on a coordinate plane.

2. Right �DJL with hypotenuse −−

DJ ; LJ = 1 − 2 DL and

−−− DL is a

units long.

3. Two of the vertices of an equilateral triangle are (0, -a) and (0, a). The third vertex is on the x-axis to the right of the origin. Name the coordinates of the third vertex.

For Questions 4 and 5, complete the coordinate proof by supplying the missing information for each corresponding location.

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 as follows: (Question 4). Since (Question 5), �ABC is isosceles.

1.

2.

3.

4.

5.

4

4



x

y

SCORE

21


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SCORE

1. Use a ruler to measure each side and a protractorto measure each angle. Which best describes the type of triangle?

A acute scalene B obtuse equilateral C acute isosceles D obtuse isosceles

Find the missing angle measures.

2. What is m∠1? F 50 H 100 G 60 J 105

3. What is m∠2? A 40 C 60 B 50 D 100

4. If �SJL � �DMT, which segment corresponds to −−

LS ? F

−−− DT H

−−− MD

G −−−

TD J −−−

MT

50°

50°

60°

2

1

5.

6.

7.

8.

1.

2.

3.

4.

Part II

5. Find the measures of the sides of �ABC with vertices A(1, 3), B(5, -2), and C(0, -4). Classify the triangle by its sides.

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

7. What is the maximum number of acute angles that a triangle can have?

8. Determine whether �DEF � �JKL, given that D(2, 0), E(5, 0), F(5, 5), J(3, -7), K(6, -7), and L(6, -2).

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

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

SCORE SCORE


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

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

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

Choose the correct term to complete the sentence.

3. If you slide, flip, or turn a triangle you are performing a (coordinate proof or congruence transformation).

4. An (interior or exterior) angle is formed by one side of a triangle and the extension of another side.

5. The SAS Postulate involves two corresponding sides and the (exterior angle or included angle) they form.

Choose from the terms above to complete each sentence.

6. A(n) organizes a series of statements in logical order, starting with the given statements.

7. A triangle that has one 90° angle is called a(n) .

8. The ASA postulate involves two corresponding angles and their corresponding .

9. A(n) uses figures in the coordinate plane and algebra to prove geometric concepts.

10. The is formed by the congruent legs of an isosceles triangle.

Define each term in your own words.

11. corollary

12. congruent triangles

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

acute triangle

auxiliary line

base angles

congruence

transformation

congruent triangles

coordinate proof

corollary

corresponding parts

equiangular triangle

equilateral triangle

exterior angle

flow proof

included angle

included side

isosceles triangle

obtuse triangle

reflection

remote interior angles

right triangle

rotation

scalene triangle

translation

vertex angle

4

?

?

?

?

?

Chapter 4 Vocabulary Test SCORE


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SCORE

80˚

A

E

CB

D

A C

B

10x - 5

6x + 37.5x

1.

2.

3.

4.

5.

6.

7.

8.

1. Which best describes the type of triangle?

A acute C obtuse B equiangular D right

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

F -8 H 1 − 2

G - 1 − 8 J 2

Use the figure for Questions 3 and 4.

3. What is m∠2? A 50 B 70 C 110 D 120

4. What is m∠4? F 10 G 60 H 100 J 120

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

C �AEB � �CBDD �ABE � �CDB

6. If �CJW � �AGS, m∠A = 50, m∠J = 45, and m∠S = 16x + 5, what is the value of x?

F 17.5 H 6

G 11.875 J 5

7. Two triangles are graphed with the following vertices: A(1, 1), B(0, 1), C(0, 0) and Q(−2, 1), R(−1, 1), S(−1, 0). When a transformation is applied to �ABC, the result is �QRS. Which best describes the transformation?

A �QRS is a translation of �ABC B �QRS is a rotation of �ABC C �QRS is a reflection of �ABC D �QRS and �ABC are not congruent

8. A triangular-shaped roof of a house has congruent legs. What is the measure of each of the two base angles?

F 25 H 100

G 50 J 120

70°

2 13 460°

40°

4 Chapter 4 Test, Form 1

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

C W

J

(16x + 5)°

45°

A S

G

50°

SCORE


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N

K

L MJ

x

yM(a, c)

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

1 2

M

TL N

Use the proof for Questions 9 and 10.

Given: L is the midpoint of −−−

JM −−

JK || −−−

NM Prove: �JKL � �MNLProof:

Statements Reasons

1. L is the midpoint of −−−

JM . 1. Given

2. −−

JL � −−−

ML 2. Definition of midpoint

3. −−

JK ‖ −−−

MN 3. Given

4. ∠JKL � ∠MNL 4. Alt. int. � are �.

5. ∠JLK � ∠MLN 5. (Question 9)

6. �JKL � �MNL 6. (Question 10)

9. What is the reason for ∠JLK � ∠MLN? A definition of midpoint B corresponding angles C vertical angles D alternate interior angles

10. What is the reason for �JKL � �MNL? F AAS G ASA H SAS J SSS


11. If �LMN is isosceles and T is the midpoint of −−−

LN , 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?

F CPCTC G definition of isosceles triangle H definition of perpendicular J definition of angle bisector

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

Bonus Classify the triangle with coordinates A(5, 0), B(0, 5), and C(-5, 0).

4 Chapter 4 Test, Form 1 (continued)

9.

10.

11.

12.

13.

B:


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SCORE


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

2. How would �ABC with vertices A(4, 1), B(2, -1), and C(-2, -1) be classified based on the length of its sides?

F equilateral G isosceles H scalene J right


3. What is m∠1? A 40 B 50 C 70 D 90

4. What is m∠3? F 40 G 70 H 90 J 110

5. If �DJL � �EGS, which segment in �EGS corresponds to −−−

DL ? A

−−− EG B

−− ES C

−−− GS D

−−− GE

6. Which triangles are congruent in the figure? F �KLJ � �MNL G �JLK � �NLM

H �JKL � �LMNJ �JKL � �MNL

7. Quadrilateral MNQP is made of two congruent triangles. −−−

NP bisects ∠N and ∠P. In the quadrilateral, m∠N = 50 and m∠P = 100. What is the measure of ∠M?

A 25 C 60 B 50 D 105

8. The coordinates of the vertices of �CDE are C(-3, 1), D(-1, 4), and E(-6, 4). A transformation applied to �CDE creates a congruent triangle �SQR. The new coordinates of two vertices are Q(-1, 6) and R(-6, 6). What are the coordinates of S?

F (-3, 3) G (1, 3) H (-1, 1) J (-1, 3)

M

N

Q

P

LK

NJ

M

70°

50°1

2

3

6x - 3

9x - 123x + 6

1.

2.

3.

4.

5.

6.

7.

8.

4 Chapter 4 Test, Form 2A SCORE


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D A

NY

A C

B

E F

x

y

B(0, a)

A(?, ?)

x

y(0, c)

(?, ?) (2b, 0)

9.

10.

11.

12.

13.

B:

9. If �ABC is isosceles with vertex angle ∠B, and

−−AE �

−−FC , which theorem or postulate can be

used to prove �AEB � �CFB? A SSS C ASA B SAS D AAS


Given: −−−

DA || −−−

YN

−−− DA �

−−− YN

Prove: ∠NDY � ∠DNAProof:

Statements Reasons

1. −−−

DA ‖ −−−

YN 1. Given

2. ∠ADN � ∠YND 2. Alt. int. � are �.

3. −−−

DA � −−−

YN 3. Given

4. −−−

DN � −−−

DN 4. Reflexive Property

5. �NDY � �DNA 5. (Question 10)

6. ∠NDY � ∠DNA 6. (Question 11)

10. What is the reason for statement 5? F ASA H SAS G AAS J SSS

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

12. What is the classification of a triangle with vertices A(3, 3), B(6, -2), C(0, -2) by the length of its sides?

F isosceles H equilateral G scalene J right

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

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

4 Chapter 4 Test, Form 2A (continued)


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SCORE


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

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

F equilateral G isosceles H scalene J right


3. What is m∠1? A 120 B 90 C 60 D 30

4. What is m∠2?

F 120 G 90 H 60 J 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? F �HMN � �HGN G �HMN � �NGH H �NMH � �NGH J �MNH � �HGN

7. The rhombus QRST is made of two congruent isosceles triangles. Given m∠QRS = 34, what is the measure of ∠S?

A 56 C 112 B 73 D 146

8. The vertices of �ABC are A(2, 0), B(5, 0), C(2, 6). The vertices of �DEF are D(-6, -7), E(-3, -7), F(-6, -1) so that �ABC � �DEF. Which best describes the congruence transformation?

F flip G rotation H reflection J translation

H G

NM

120°12

7x - 15

3x + 54x

1.

2.

3.

4.

5.

6.

7.

8.

4 Chapter 4 Test, Form 2B

Q

T S

R


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G A

N

E I

A DF E

CB

(5x + 60)° (2x + 51)°

(30 - 10x)°(43 - 2x)°

x

y

(?, ?)

(a, 0)(-a, 0)

9.

10.

11.

12.

13.

B:

9. If −−AF �

−−−DE ,

−−AB �

−−FC and

−−AB ||

−−FC , which theorem

or postulate can be used to prove �ABE � �FCD? A AAS C SAS B ASA D SSS


Given: −−−

EG � −−

IA ∠EGA � ∠IAG

Prove: ∠GEN � ∠AINProof:

Statements Reasons

1. −−−

EG � −−

IA 1. Given

2. ∠EGA � ∠IAG 2. Given

3. −−−

GA � −−−

GA 3. Reflexive Property

4. �EGA � �IAG 4. (Question 10)

5. ∠GEN � ∠AIN 5. (Question 11)

10. What is the reason for statement 4? F SSS G ASA H SAS J AAS

11. What is the reason for statement 5? A Alt. int. � are �. C Corr. angles are �. B Same side interior angles D CPCTC

12. What is the classification of a triangle with vertices A(-3, -1), B(-2, 2), C(3, 1) by its sides?

F scalene H equilateral G isosceles J right

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

2 )

Bonus Find the value of x.

4 Chapter 4 Test, Form 2B (continued)


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SCORE

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 the triangle by its sides.

Use the figure for Question 4-6. Find the measure of each angle.

4. m∠1

5. m∠2

6. m∠3

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

8. Identify the transformation and verify that it is a congruence transformation.

x

y

A

D F

GB

C

110° 2

1

3

A C

B

8x

7x + 310x - 6

1.

2.

3.

4.

5.

6.

7.

8.

4 Chapter 4 Test, Form 2C


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9. A rectangular piece of paper measuring 8 inches by 16 inches is folded in half to form a square. It is then folded again along its diagonal to form a triangle. Find the measures of the angles of the triangle.

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

11. Find m∠1.


13. Position and label isosceles �ABC with base −−

AB , b units long, on the coordinate plane.

14. −−

CP joins point C in isosceles right �ABC to the midpoint P, of

−− AB .

Name the coordinates of P. Then determine the relationship between

−− AB and

−− CP .

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

A

B

C D

E

F X

Y

Z M

N

O

x

y

A(0, b)

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

P

(10x + 20)°

15x°

(18x - 12)°

40°

1

P

1 2

N MK

L

4 Chapter 4 Test, Form 2C (continued)

9.

10.

11.

12.

13.

14.

B:


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SCORE

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 with AB = BC.

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 the triangle by its sides.

Use the figure for Questions 4-6. Find the measure of each angle.

4. m∠1

5. m∠2

6. m∠3

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

8. Identify the transformation and verify that it is a congruence transformation.

9. A square piece of paper measuring 8 inches on each side is folded in half forming a triangle. It is then folded in half again to form another triangle. Find the angles of the resulting triangle.

DB

EC

FA

70°1 32

80°

A C

B

2x + 20

9x - 5

5x + 5

4 Chapter 4 Test, Form 2D

x

y

1.

2.

3.

4.

5.

6.

7.

8.

9.


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10. �ABC is an isosceles triangle with −−−

BD ⊥ −−

AC . 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. Find m∠1.


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

14. −−−

MN joins the midpoint of −−

AB and the midpoint of

−− AC in �ABC. Find the

coordinates of M and N.

Bonus Without finding any other angles or sides congruent, determine 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

10.

11.

12.

13.

14.

B:

4 Chapter 4 Test, Form 2D (continued)


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SCORE

1. If �ABC is isosceles, ∠B is the vertex angle, AB = 20x - 2, BC = 12x + 30, and AC = 25x, find x and the length of each side of the triangle.

2. Find the length of the sides of the triangle with vertices A(0, 4), B(5, 4), and C(-3, -2). Classify the triangle by its sides and angles.

Use the figure to answer Questions 3–5.


4. Find m∠1, if m∠1 = 4x + 10.

5. Find m∠2.

6. The vertices of �ABC are A(-1, 1), B(4, 2), and C(1, 5). The vertices of �DEF are D(-1, -1), E(4, -2), and F(1, -5) so that �ABC � �DEF. Identify the congruence transformation.

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, −−

AC � −−−

FD , −−

AB || −−−

DE , and

−− AC ||

−−− FD . Determine which

postulate can be used to prove �ABC � �DEF. 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.

4 Chapter 4 Test, Form 3


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

Given: �ABC is an isosceles triangle with base −−

AC . D is the midpoint of

−− AC .

Prove: −−−

BD bisects ∠ABC

Proof:

Statements Reasons 1. �ABC is isosceles with

base −−

AC . 1. Given

2. −−

AB � −−−

CB 2. Def. of isosceles triangle.

3. ∠A � ∠C 3. (Question 9)

4. D is the midpoint of −−

AC . 4. Given

5. −−−

AD � −−−

CD 5. Midpoint Theorem

6. �ABD � �CBD 6. (Question 10)

7. ∠1 � ∠2 7. CPCTC

8. −−−

BD bisects ∠ABC 8. Def. of angle bisector

11. Find x.

12. Position and label isosceles �ABC with base −−

AB , (a + b) units long, on a coordinate plane

13. A wood column is supported by four cablesthat form four triangles. Determine which postulate can be used to show that all the triangles are congruent.

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

987

1

65

23

4

B

C

DE

A

(15x + 15)°(21x - 3)°

(17x + 9)°

21

A C

B

D

9.

10.

11.

12.

13.

B:

4 Chapter 4 Test, Form 3 (continued)


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SCORE

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

1.

a. 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. Determine which theorem or postulate can be used to prove that the triangles are congruent.

b. List their corresponding congruent angles and sides.

5. Write a two-column proof.

Given: −−

AB || −−−

DE , −−−

AD bisects −−−

BE . Prove: �ABC � �DEC 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)°

4 Chapter 4 Extended-Response Test


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4 Standardized Test Practice(Chapters 1– 4)

A CB

D

?

U V

XW

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

A x = 2, y = 12 C x = 12, y = 2 B x = 27, y = 1 −

3 D x = 1 −

3 , y = 27

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

F G H J

3. Complete the statement so that its conditional and its converse are true.

If ∠1 � ∠2, then ∠1 and ∠2 . (Lesson 2-3)

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

angles. 4. Complete this proof. (Lesson 2-7) Given:

−−− UV �

−−− VW

−−−

VW � −−−

WX Prove: UV = WX

Proof:

Statements Reasons

1. −−−

UV � −−−

VW ; −−−

VW � −−−

WX 1. Given

2. UV = VW; VW = WX 2.

3. UV = WX 3. Transitive Property

F Definition of congruent segments G Substitution Property H Segment Addition Postulate J Symmetric Property

5. Which equation has a slope of 1 − 3 and a y-intercept of -2? (Lesson 3-4)

A y = 1 − 3 x + 2 C y = 1 −

3 x - 2

B y = 2x - 1 − 3 D y = -2x + 1 −

3

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

F acute G equiangular H obtuse J right

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

A SAS C SSSB ASA D AAS

?

SCORE

Part 1: Multiple Choice

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

1.

2. F G H J

3. A B C D

4. F G H J

5.

6. F G H J

7. A B C D

A B C D

A B C D


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SCORE 4 Standardized Test Practice (continued)

8. If the volume of a cylinder with a height of 3 feet is 75π cubic feet, find the surface area of the cylinder in square feet. (Lesson 1-7)

F 25π G 50π H 80π J 30π

9. Let C be the midpoint of the line segment AB having endpoints A(2, -7) and B(6, -3). Find the length of

−−− CB . (Lesson 1-3)

A 2 √ � 2 units B √ �� 53 units C √ �� 20 units D √ �� 10 units

10. Find x so that p || q. (Lesson 3-5)

F 11.50 H 20.5 G 20 J 25

11. If ∠1 is supplementary to ∠2 and ∠3 is complementary to ∠2, find m∠3 if m∠1 is 145. (Lesson 2-8)

A 35 C 55 B 45 D 90

12. Let �ABC be an isosceles triangle with �ABC � �PQR. If m∠B = 154, find m∠R. (Lesson 4-3)

F 154 G 126 H 26 J 13

13. In proving the congruence of two triangles, which postulate involves an included angle? (Lessons 4-4 and 4-5)

A SSS B SAS C ASA D AAS

14. �PQR is an isosceles triangle with base −−−

QR . If m∠P = 6x + 40 and m∠Q = x - 10, find x. (Lesson 4-6)

F 20 G 25 H 30 J 100

qp(2x

- 55

)°

(8x + 30)°

8.

9.

10.

11.

12.

13.

14. F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

15. If −−

JK || −−− LM , then ∠4 must

be supplementary to ∠ . (Lesson 3-5)

16. Find PR if �PQR is isosceles, ∠Q is the vertex angle, PQ = 4x - 8, QR = x + 7, and PR = 6x - 12. (Lesson 4-1)

?

6

7

4

5H

L

M

J

K

15.

16. 9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

Part 2: Gridded ResponsePart 2: Gridded Response

Instructions: Enter your answer by writing each digit of the

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

that corresponds to that entry.


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17. The perimeter of a regular pentagon is 14.5 feet. Find the new perimeter if the length of each side of the pentagon is doubled. (Lesson 1-6)

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

19. Find m∠PQR. (Lesson 4-2)

20. If PQ = QS, QS = SR, and m∠R = 20, find m∠PSQ. (Lesson 4-6)

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

22. Refer to the figure at the right to answer the questions below.

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

b. Classify �ADC. (Lesson 4-1)

c. Find m∠ACD. (Lesson 4-2)

50°30°

85°ED

A CB

F

P RS

Q

63° 125°10°

P S

Q

RT

17.

18.

19.

20.

21.

22a.

22b.

22c.

4 Standardized Test Practice (continued)

Part 3: Short Response

Instructions: Write your answers in the space provided.


An

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

NA

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Lesson 4-1

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lass

ify

each

tri

angl

e as

acu

te, e

qu

ian

gula

r, o

btu

se, o

r ri

ght.

1.

67°

90°

23°

K LM

2.

120°

30°

30°

NO

P

3.

60°

60°

60°

Q

RS

r

igh

t o

btu

se

eq

uia

ng

ula

r

4.

65°

65°

50°

UV

T

5.

45°

45°

90°

XY

W

6.

60°

28°

92°

FDB

a

cu

te

rig

ht

ob

tuse

4-1

Exam

ple

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Chapter Resources


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

3

Gle

ncoe

Geo

met

ry

Ant

icip

atio

n G

uide

Co

ng

ruen

t Tri

an

gle

s

B

efor

e yo

u b

egin

Ch

ap

ter

4

• R

ead

each

sta

tem

ent.

• D

ecid

e w

het

her

you

Agr

ee (

A)

or D

isag

ree

(D)

wit

h t

he

stat

emen

t.

• W

rite

A o

r D

in

th

e fi

rst

colu

mn

OR

if

you

are

not

su

re w

het

her

you

agr

ee o

r di

sagr

ee,

wri

te N

S (

Not

Su

re).

A

fter

you

com

ple

te C

ha

pte

r 4

• R

erea

d ea

ch s

tate

men

t an

d co

mpl

ete

the

last

col

um

n b

y en

teri

ng

an A

or

a D

.

• D

id a

ny

of y

our

opin

ion

s ab

out

the

stat

emen

ts c

han

ge f

rom

th

e fi

rst

colu

mn

?

• F

or t

hos

e st

atem

ents

th

at y

ou m

ark

wit

h a

D, u

se a

pie

ce o

f pa

per

to w

rite

an

ex

ampl

e of

wh

y yo

u d

isag

ree.

4 Step

1

ST

EP

1A

, D

, o

r N

SS

tate

men

tS

TE

P 2

A o

r D

1.

For

a t

rian

gle

to b

e ac

ute

, all

th

ree

angl

es m

ust

be

acu

te.

A

2.

All

sid

es o

f an

equ

ilat

eral

tri

angl

e ar

e co

ngr

uen

t.A

3.

An

ext

erio

r an

gle

is a

ny

angl

e ou

tsid

e of

th

e gi

ven

tr

ian

gle.

D

4.

Th

e su

m o

f th

e m

easu

res

of t

he

angl

es i

n a

tri

angl

e is

360

°.D

5.

An

obt

use

tri

angl

e is

a t

rian

gle

wit

h t

hre

e ob

tuse

an

gles

.D

6.

Tri

angl

es t

hat

are

th

e sa

me

shap

e an

d si

ze a

re

con

gru

ent.

A

7.

If t

he

angl

es o

f on

e tr

ian

gle

are

con

gru

ent

to t

he

angl

es o

f an

oth

er t

rian

gle,

th

en t

he

two

tria

ngl

es a

re

con

gru

ent.

D

8.

An

iso

scel

es t

rian

gle

is a

tri

angl

e w

ith

tw

o co

ngr

uen

t si

des.

A

9.

A t

rian

gle

can

be

equ

ian

gula

r an

d n

ot e

quil

ater

al.

D

10.

To

mak

e th

e ca

lcu

lati

ons

in a

coo

rdin

ate

proo

f ea

sier

, pl

ace

eith

er a

ver

tex

or t

he

cen

ter

of t

he

poly

gon

at

the

orig

in.

A

Step

2

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Answers (Anticipation Guide and Lesson 4-1)

Chapter 4 A1 Glencoe Geometry

A01_A26_GEOCRM04_890513.indd 1A01_A26_GEOCRM04_890513.indd 1 5/23/08 11:38:07 PM5/23/08 11:38:07 PM

Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass

Lesson 4-1


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

7

Gle

ncoe

Geo

met

ry

Skill

s Pr

acti

ceC

lassif

yin

g T

rian

gle

s

Cla

ssif

y ea

ch t

rian

gle

as a

cute

, eq

uia

ngu

lar,

obt

use

, or

righ

t.

1.

60°

60°

60°

2.

40

° 95°

45°

3.

50°

40°

90°

e

qu

ian

gu

lar

ob

tuse

rig

ht

4.

80°

50°

50°

5.

100°

30°

50°

6.

70

°

55°

55°

a

cu

te

ob

tuse

acu

te

Cla

ssif

y ea

ch t

rian

gle

as e

qu

ila

tera

l, i

sosc

eles

, or

sca

len

e.

EC

D

AB

9

88√2

7. �

AB

E

8. �

ED

B

s

cale

ne

iso

scele

s

9. �

EB

C

10. �

DB

C

s

cale

ne

eq

uala

tera

l

11. A

LGEB

RA

Fin

d x

and

the

len

gth

of

each

sid

e if

�A

BC

is

an i

sosc

eles

tr

ian

gle

wit

h −−

A

B �

−−

BC

.

12. A

LGEB

RA

Fin

d x

and

the

len

gth

of

each

si

de i

f �

FG

H i

s an

equ

ilat

eral

tri

angl

e.

5x-

83x

+10

4x+

1

13. A

LGEB

RA

Fin

d x

and

the

len

gth

of

each

sid

e if

�R

ST

is

an i

sosc

eles

tr

ian

gle

wit

h −

−

RS

� −

−

TS .

3x-

1

9x+

2

6x+

8

14. A

LGEB

RA

Fin

d x

and

the

len

gth

of

each

si

de i

f �

DE

F i

s an

equ

ilat

eral

tri

angl

e.

5x-

212x

+6

7x-

39

4-1 x =

4;

AB

=

B

C =

13;

AC

= 6

x =

9;

FG

=

G

H =

FH

= 3

7

x =

2;

RS

=

S

T =

20;

RT

= 5

x =

9;

DE

=

E

F =

DF =

24

4x-

32x

+5

x+

2

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NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

6

Gle

ncoe

Geo

met

ry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Cla

ssif

yin

g T

rian

gle

s

Cla

ssif

y Tr

ian

gle

s b

y Si

des

You

can

cla

ssif

y a

tria

ngl

e by

th

e n

um

ber

of c

ongr

uen

t si

des.

Equ

al n

um

bers

of

has

h m

arks

in

dica

te c

ongr

uen

t si

des.

• If

all

thre

e sid

es o

f a

tria

ng

le a

re c

on

gru

en

t, t

he

n t

he

tria

ng

le is a

n e

qu

ila

tera

l tr

ian

gle

.

• If

at

leas

t tw

o sid

es o

f a

tria

ng

le a

re c

on

gru

en

t, t

he

n t

he

tria

ng

le is a

n i

so

sc

ele

s t

ria

ng

le.

Eq

uila

tera

l tr

ian

gle

s c

an

als

o b

e c

on

sid

ere

d iso

sce

les.

• If

no

two

sid

es o

f a

tria

ng

le a

re c

on

gru

en

t, t

he

n t

he

tria

ng

le is a

sc

ale

ne

tri

an

gle

.

C

lass

ify

each

tri

angl

e.

a. L

J

H

b.

N

RP

c.

23

12

15X

V

T

T

wo

side

s ar

e co

ngr

uen

t.

All

th

ree

side

s ar

e

Th

e tr

ian

gle

has

no

pair

Th

e tr

ian

gle

is a

n

con

gru

ent.

Th

e tr

ian

gle

o

f co

ngr

uen

t si

des.

It

is

isos

cele

s tr

ian

gle.

i

s an

equ

ilat

eral

tri

angl

e.

a s

cale

ne

tria

ngl

e.

Exer

cise

sC

lass

ify

each

tri

angl

e as

eq

uil

ate

ral,

iso

scel

es, o

r sc

ale

ne.

1.

21

3√

⎯G

CA

2.

G

KI

18

1818

3.

12

17

19Q

O

M

s

cale

ne

eq

uilate

ral

scale

ne

4.

UW

S

5. 8x

32x

32x

B CA

6.

DE

Fx

xx

i

so

scele

s

iso

scele

s

eq

uilate

ral

Exam

ple

4-1

7. A

LGEB

RA

Fin

d x

and

the

len

gth

of

each

si

de i

f �

RS

T i

s an

equ

ilat

eral

tri

angl

e. 8

. ALG

EBR

A F

ind

x an

d th

e le

ngt

h o

f ea

ch s

ide

if �

AB

C i

s is

osce

les

wit

h

AB

= B

C.

5x-

42x

+2

3x

3y+

24y

3y

2;

62;

AB

= 8;

BC

= 8, A

C =

6

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:34:

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M

Answers (Lesson 4-1)



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

8

Gle

ncoe

Geo

met

ry

Prac

tice

Cla

ssif

yin

g T

rian

gle

s

Cla

ssif

y ea

ch t

rian

gle

as a

cute

, eq

uia

ngu

lar,

obt

use

, or

righ

t.

1.

2.

3.

Cla

ssif

y ea

ch t

rian

gle

in t

he

figu

re a

t th

e ri

ght

by

its

angl

es a

nd

sid

es.

4. �

AB

D

5. �

AB

C

e

qu

ian

gu

lar;

eq

uilate

ral

rig

ht;

scale

ne

6. �

ED

C

7. �

BD

C

r

igh

t; s

cale

ne

ob

tuse;

iso

scele

s

ALG

EBR

A F

or e

ach

tri

angl

e, f

ind

x a

nd

th

e m

easu

re o

f ea

ch s

ide.

8. �

FG

H i

s an

equ

ilat

eral

tri

angl

e w

ith

FG

= x

+ 5

, GH

= 3

x -

9, a

nd

FH

= 2

x -

2.

9. �

LM

N i

s an

iso

scel

es t

rian

gle,

wit

h L

M =

LN

, LM

= 3

x -

2, L

N =

2x

+ 1

, an

d M

N =

5x

- 2

.

Fin

d t

he

mea

sure

s of

th

e si

des

of

�K

PL

an

d c

lass

ify

each

tri

angl

e b

y it

s si

des

.

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. D

ESIG

N D

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.

How

man

y ri

ght

angl

es a

re t

her

e?

AC

DE

B

60°

30°

90°

65°

30°

85°

40°

100°

40°

4-1 x =

7;

FG

=

12,

GH

= 12,

FH

= 1

2

x =

3;

LM

=

7,

LN

= 7,

MN

= 1

3

KP

= √

��

26 ,

PL

=

4 √

�

2 ,

LK

= √

��

26 ;

iso

scele

s

KP

= √

��

53 ,

PL

=

5,

LK

= 2 √

��

13 ; s

cale

ne

KP

= 2

√ �

�

10 ,

PL

=

5 √

�

2 ,

LK

= 5 √

�

2 ;

iso

scele

s

5

o

btu

se

acu

te

rig

ht

001_

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:34:

17A

M

Lesson 4-1


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

9

Gle

ncoe

Geo

met

ry

Wor

d Pr

oble

m P

ract

ice

Cla

ssif

yin

g T

rian

gle

s

1. M

USE

UM

SP

aul

is s

tan

din

g in

fro

nt

of

a m

use

um

exh

ibit

ion

. Wh

en h

e tu

rns

his

h

ead

60°

to t

he

left

, he

can

see

a s

tatu

e by

Don

atel

lo. W

hen

he

turn

s h

is h

ead

60°

to t

he

righ

t, h

e ca

n s

ee a

sta

tue

by

Del

la R

obbi

a. T

he

two

stat

ues

an

d P

aul

form

th

e ve

rtic

es o

f a

tria

ngl

e. C

lass

ify

this

tri

angl

e as

acu

te, r

igh

t, o

r ob

tuse

.

2. P

APE

R M

arsh

a cu

ts a

rec

tan

gula

r pi

ece

of p

aper

in

hal

f al

ong

a di

agon

al. T

he

resu

lt i

s tw

o tr

ian

gles

. Cla

ssif

y th

ese

tria

ngl

es a

s ac

ute

, rig

ht,

or

obtu

se.

3. W

ATE

RSK

IING

Kim

an

d C

assa

ndr

a ar

e w

ater

skii

ng.

Th

ey a

re h

oldi

ng

on t

o ro

pes

that

are

th

e sa

me

len

gth

an

d ti

ed

to t

he

sam

e po

int

on t

he

back

of

a sp

eed

boat

. Th

e bo

at i

s go

ing

full

spe

ed a

hea

d an

d th

e ro

pes

are

full

y ta

ut.

K

im, C

assa

ndr

a, a

nd

the

poin

t w

her

e th

e ro

pes

are

tied

on

th

e bo

at f

orm

th

e ve

rtic

es o

f a

tria

ngl

e. T

he

dist

ance

be

twee

n K

im a

nd

Cas

san

dra

is n

ever

eq

ual

to

the

len

gth

of

the

rope

s. C

lass

ify

the

tria

ngl

e as

equ

ilat

eral

, iso

scel

es, o

r sc

alen

e.

4. B

OO

KEN

DS

Tw

o bo

oken

ds a

re s

hap

ed

like

rig

ht

tria

ngl

es.

Th

e bo

ttom

sid

e of

eac

h t

rian

gle

is

exac

tly

hal

f as

lon

g as

th

e sl

ante

d si

de

of t

he

tria

ngl

e. I

f al

l th

e bo

oks

betw

een

th

e bo

oken

ds a

re r

emov

ed a

nd

they

are

pu

shed

tog

eth

er, t

hey

wil

l fo

rm a

sin

gle

tria

ngl

e. C

lass

ify

the

tria

ngl

e th

at c

an

be f

orm

ed a

s eq

uil

ater

al, i

sosc

eles

, or

scal

ene.

5. D

ESIG

NS

Su

zan

ne

saw

th

is p

atte

rn o

n

a pe

nta

gon

al f

loor

til

e. S

he

not

iced

man

y di

ffer

ent

kin

ds o

f tr

ian

gles

wer

e cr

eate

d by

th

e li

nes

on

th

e ti

le.

a. I

den

tify

fiv

e tr

ian

gles

th

at a

ppea

r to

be

acu

te i

sosc

eles

tri

angl

es.

b.

Iden

tify

fiv

e tr

ian

gles

th

at a

ppea

r to

be

obt

use

iso

scel

es t

rian

gles

.

4-1

Kim

Cass

andr

a

CA

GH

IF

JB

ED

Th

ey a

re b

oth

rig

ht

tria

ng

les.

ob

tuse

eq

uilate

ral

Sam

ple

an

sw

ers

: �

EB

D,

�C

AE

, �

AD

C,

�B

GH

, �

CH

I, �

DIJ

, �

EJF,

�A

GF

Sam

ple

an

sw

ers

: �

AFE

, �

EJD

, �

CID

, �

BH

C,

�B

GA

, �

AH

D,

�C

GE

, �

AC

J

iso

scele

s

001_

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:34:

26A

M




Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

10

G

lenc

oe G

eom

etry

Enri

chm

ent

Read

ing

Math

em

ati

cs

Wh

en y

ou r

ead

geom

etry

, you

may

nee

d to

dra

w a

dia

gram

to

mak

e th

e te

xt

easi

er t

o u

nde

rsta

nd.

C

onsi

der

th

ree

poi

nts

, A, B

, an

d C

on

a c

oord

inat

e gr

id. T

he

y-co

ord

inat

es o

f A

an

d B

are

th

e sa

me.

Th

e x-

coor

din

ate

of B

is

grea

ter

than

th

e x-

coor

din

ate

of A

. Bot

h c

oord

inat

es o

f C

are

gre

ater

th

an t

he

corr

esp

ond

ing

coor

din

ates

of

B. I

s �

AB

C a

cute

, rig

ht,

or

obtu

se?

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 −

−

AB

mu

st b

e a

hor

izon

tal

segm

ent

beca

use

th

e y-

coor

din

ates

are

th

e sa

me.

Poi

nt

C m

ust

be

loca

ted

to t

he

righ

t an

d u

p fr

om p

oin

t B

.F

rom

th

e di

agra

m y

ou c

an s

ee t

hat

tri

angl

e A

BC

m

ust

be

obtu

se.

Exer

cise

sD

raw

a s

imp

le t

rian

gle

on g

rid

pap

er t

o h

elp

you

.

1. C

onsi

der

thre

e po

ints

, R, S

, an

d

2. C

onsi

der

thre

e n

onco

llin

ear

poin

ts,

T o

n a

coo

rdin

ate

grid

. Th

e

J, K

, an

d L

on

a c

oord

inat

e gr

id. T

he

x-co

ordi

nat

es o

f R

an

d S

are

th

e y

-coo

rdin

ates

of

J a

nd

K a

re t

he

sam

e. T

he

y-co

ordi

nat

e of

T i

s s

ame.

Th

e x-

coor

din

ates

of

K a

nd

Lbe

twee

n t

he

y-co

ordi

nat

es o

f R

a

re t

he

sam

e. I

s �

JK

L a

cute

,an

d S

. Th

e x-

coor

din

ate

of T

is

less

r

igh

t, o

r ob

tuse

? th

an t

he

x-co

ordi

nat

e of

R. I

s an

gle

R

of

�R

ST

acu

te, r

igh

t, o

r ob

tuse

?

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 c

oord

inat

e gr

id.

on

a c

oord

inat

e gr

id. P

oin

ts G

an

d T

he

x-co

ordi

nat

es o

f D

an

d E

are

H

are

on

th

e po

siti

ve y

-axi

s, a

nd

oppo

site

s. T

he

y-co

ordi

nat

es o

f D

an

d t

he

y-co

ordi

nat

e of

G i

s tw

ice

the

E a

re t

he

sam

e. T

he

x-co

ordi

nat

e of

y

-coo

rdin

ate

of H

. Poi

nt

I is

on

th

e F

is

0. W

hat

kin

d of

tri

angl

e m

ust

p

osit

ive

x-ax

is, a

nd

the

x-co

ordi

nat

e�

DE

F b

e: s

cale

ne,

iso

scel

es, o

r o

f I

is g

reat

er t

han

th

e y-

coor

din

ate

equ

ilat

eral

?

of

G. I

s �

GH

I sc

alen

e, i

sosc

eles

, or

equ

ilat

eral

?

BA

C x

y

O

4-1

Exam

ple

iso

scele

s

rig

ht

scale

ne

acu

te

001_

018_

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OC

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812

:34:

31A

M

Lesson 4-2


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

11

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

An

gle

s o

f Tri

an

gle

s

Tria

ng

le A

ng

le-S

um

Th

eore

m I

f th

e m

easu

res

of t

wo

angl

es o

f a

tria

ngl

e ar

e kn

own

, th

e m

easu

re o

f th

e th

ird

angl

e ca

n a

lway

s be

fou

nd.

Tri

an

gle

An

gle

Su

m

Th

eo

rem

Th

e s

um

of

the

me

asu

res o

f th

e a

ng

les o

f a

tria

ng

le is 1

80

.

In t

he

fig

ure

at

the

rig

ht,

m∠

A +

m∠

B +

m∠

C =

18

0.

CA

B

F

ind

m∠

T.

m∠

R +

m∠

S +

m∠

T =

180

T

riangle

Angle

-

Sum

Theore

m

25

+ 3

5 +

m∠

T =

180

S

ubstitu

tion

60

+ m

∠T

= 1

80

Sim

plif

y.

m

∠T

= 1

20

Subtr

act

60 f

rom

each s

ide.

F

ind

th

e m

issi

ng

angl

e m

easu

res.

m∠

1 +

m∠

A +

m∠

B =

180

T

riangle

Angle

- S

um

T

heore

m

m∠

1 +

58

+ 9

0 =

180

S

ubstitu

tion

m∠

1 +

148

= 1

80

Sim

plif

y.

m∠

1 =

32

Subtr

act

148 f

rom

each s

ide.

m∠

2 =

32

Vert

ical angle

s a

re

congru

ent.

m∠

3 +

m∠

2 +

m∠

E =

180

T

riangle

Angle

- S

um

T

heore

m

m∠

3 +

32

+ 1

08 =

180

S

ubstitu

tion

m∠

3 +

140

= 1

80

Sim

plif

y.

m∠

3 =

40

Subtr

act

140 f

rom

each s

ide.

58°90

°

108°

12

3

E

DA

C

B

Exer

cise

sF

ind

th

e m

easu

res

of e

ach

nu

mb

ered

an

gle.

1.

2.

3.

4.

5.

6.

20°

152°

DG

A

130

°60

°

12

S

R

TW

V WT

U

30°

60°

2

1

S

QR

30°

1

90°

62°

1NM

P4-2

Exam

ple

1

35°

25°

RT

S

Exam

ple

2

m∠

1 =

28

m∠

1 =

120

m∠

1 =

30,

m∠

2 =

60

m∠

1 =

56,

m∠

2 =

56,

m∠

3 =

74

m∠

1 =

30,

m∠

2 =

60

m∠

1 =

8

Q

O

NM P

58°

66°

50°

32

1

001_

018_

GE

OC

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:34:

35A

M

Answers (Lesson 4-1 and Lesson 4-2)



An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

12

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

An

gle

s o

f Tri

an

gle

s

Exte

rio

r A

ng

le T

heo

rem

At

each

ver

tex

of a

tri

angl

e, t

he

angl

e fo

rmed

by

one

side

an

d an

ext

ensi

on o

f th

e ot

her

sid

e is

cal

led

an e

xter

ior

angl

e of

th

e tr

ian

gle.

For

eac

h

exte

rior

an

gle

of a

tri

angl

e, t

he

rem

ote

inte

rior

an

gles

are

th

e in

teri

or a

ngl

es t

hat

are

not

ad

jace

nt

to t

hat

ext

erio

r an

gle.

In

th

e di

agra

m b

elow

, ∠B

an

d ∠

A a

re t

he

rem

ote

inte

rior

an

gles

for

ext

erio

r ∠

DC

B.

Ex

teri

or

An

gle

Th

eo

rem

Th

e m

ea

su

re o

f a

n e

xte

rio

r a

ng

le o

f a

tria

ng

le is e

qu

al to

the

su

m o

f th

e m

ea

su

res o

f th

e t

wo

re

mo

te in

terio

r a

ng

les.

AC

B

D1

m∠

1 =

m∠

A +

m∠

B

F

ind

m∠

1.

m∠

1 =

m∠

R +

m∠

S

Exte

rior

Angle

Theore

m

= 6

0 +

80

Substitu

tion

= 1

40

Sim

plif

y.

RT

S

60°

80°

1

F

ind

x.

m

∠P

QS

= m

∠R

+ m

∠S

E

xte

rior

Angle

Theore

m

78

= 5

5 +

x

Substitu

tion

23

= x

S

ubtr

act

55 f

rom

each

sid

e.

SR

Q

P

55°

78°

x°

Exer

cise

s

Fin

d t

he

mea

sure

s of

eac

h n

um

ber

ed a

ngl

e.

1.

2.

3.

4.

Fin

d e

ach

mea

sure

.

5. m

∠A

BC

6.

m∠

FE

FG

H58

°x°

x°B

A

DC

95°

2x°

145°

UT

SR

V

35°

36°

80°

13

2

PO

Q

NM

60°

60°

32

1

BC

D

A

25°

35° 1

2Y

ZW

X 65°50

°

1

4-2

Exam

ple

1Ex

amp

le 2

m∠

1 =

115

m∠

1 =

60,

m∠

2 =

120

m∠

1 =

60,

m∠

2 =

60,

m∠

3 =

12

0m

∠1 =

109,

m∠

2 =

29,

m∠

3 =

71

50

29

001_

018_

GE

OC

RM

C04

_890

513.

indd

124/

12/0

812

:34:

41A

M

Lesson 4-2


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

13

G

lenc

oe G

eom

etry

Skill

s Pr

acti

ceA

ng

les o

f Tri

an

gle

s

Fin

d t

he

mea

sure

of

each

nu

mb

ered

an

gle.

1.

2.

Fin

d e

ach

mea

sure

.

3. m

∠1

4. m

∠2

5. m

∠3

Fin

d e

ach

mea

sure

.

6. m

∠1

7. m

∠2

8. m

∠3

Fin

d e

ach

mea

sure

.

9. m

∠1

10. m

∠2

11. m

∠3

12. m

∠4

13. m

∠5

Fin

d e

ach

mea

sure

.

14. m

∠1

15. m

∠2

63°

1

2D

AC

B

80°

60°

40°

105°

14

52

3

150°

55°

70°

12

3

85°

55°

40°

12

3

146°

12

TIG

ERS

80° 73°

1

4-2 m

∠1 =

27

m∠

1 =

m∠

2 =

17

55

55

70

125

55

95

140

40

65

75

115

27

27

001_

018_

GE

OC

RM

C04

_890

513.

indd

135/

22/0

88:

04:0

2P

M




Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

14

G

lenc

oe G

eom

etry

Prac

tice

An

gle

s o

f Tri

an

gle

s

Fin

d t

he

mea

sure

of

each

nu

mb

ered

an

gle.

1.

2.

Fin

d e

ach

mea

sure

.

3. m

∠1

97

4. m

∠2

83

5. m

∠3

62

Fin

d e

ach

mea

sure

.

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 e

ach

mea

sure

.

12. m

∠1

26

13. m

∠2

32

14. C

ON

STR

UC

TIO

N T

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.

145°

1

64°

58°

1

2AB

C

D

118°

36°

68°

70° 65

° 82°

1

2

34

5

6

58°

39°

35°

12

3

40°

1

55°

72°

1

2

4-2 55

m

∠1 =

18,

m∠

1 =

90

m∠

1 =

85

001_

018_

GE

OC

RM

C04

_890

513.

indd

144/

12/0

812

:34:

53A

M

Lesson 4-2


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

15

G

lenc

oe G

eom

etry

Wor

d Pr

oble

m P

ract

ice

An

gle

s o

f Tri

an

gle

s

62˚

136

˚

1. P

ATH

S E

ric

wal

ks a

rou

nd

a tr

ian

gula

r pa

th. A

t ea

ch c

orn

er, h

e re

cord

s th

e m

easu

re o

f th

e an

gle

he

crea

tes.

He

mak

es o

ne

com

plet

e ci

rcu

it a

rou

nd

the

path

. Wh

at i

s th

e su

m o

f th

e th

ree

angl

e m

easu

res

that

he

wro

te

dow

n?

2. S

TAN

DIN

G S

am, K

endr

a, a

nd

Ton

y ar

e st

andi

ng

in s

uch

a w

ay t

hat

if

lin

es w

ere

draw

n t

o co

nn

ect

the

frie

nds

th

ey w

ould

fo

rm a

tri

angl

e.

If S

am i

s lo

okin

g at

Ken

dra

he

nee

ds t

o tu

rn h

is h

ead

40°

to l

ook

at T

ony.

If

Ton

y is

loo

kin

g at

Sam

he

nee

ds t

o tu

rn

his

hea

d 50

° to

loo

k at

Ken

dra.

How

m

any

degr

ees

wou

ld K

endr

a h

ave

to

turn

her

hea

d to

loo

k at

Ton

y if

sh

e is

lo

okin

g at

Sam

?

3. T

OW

ERS

A l

ooko

ut

tow

er s

its

on a

n

etw

ork

of s

tru

ts a

nd

post

s. L

esli

e m

easu

red

two

angl

es o

n t

he

tow

er.

W

hat

is

the

mea

sure

of

angl

e 1?

4. Z

OO

S T

he

zoo

ligh

ts u

p th

e ch

impa

nze

e pe

n w

ith

an

ove

rhea

d li

ght

at n

igh

t. T

he

cros

s se

ctio

n o

f th

e li

ght

beam

mak

es a

n

isos

cele

s tr

ian

gle.

Th

e to

p an

gle

of t

he

tria

ngl

e is

52

and

the

exte

rior

an

gle

is 1

16. W

hat

is

the

mea

sure

of

angl

e 1?

5. D

RA

FTIN

G C

hlo

e bo

ugh

t a

draf

tin

g ta

ble

and

set

it u

p so

th

at s

he

can

dra

w

com

fort

ably

fro

m h

er s

tool

. Ch

loe

mea

sure

d th

e tw

o an

gles

cre

ated

by

the

legs

an

d th

e ta

blet

op i

n c

ase

she

had

to

dism

antl

e th

e ta

ble.

a. W

hic

h o

f th

e fo

ur

nu

mbe

red

angl

es

can

Ch

loe

dete

rmin

e by

kn

owin

g th

e tw

o an

gles

for

med

wit

h t

he

tabl

etop

? W

hat

are

th

eir

mea

sure

s?

b.

Wh

at c

oncl

usi

on c

an C

hlo

e m

ake

abou

t th

e u

nkn

own

an

gles

bef

ore

she

mea

sure

s th

em t

o fi

nd

thei

r ex

act

mea

sure

men

ts?

4-2

52˚

116 ˚

1

Kend

ra

Tony

Sam

29˚

68˚

12

34

180

90

98

64

∠1 a

nd

∠2;

m∠

1 =

97,

m∠

2 =

83

Th

e s

um

of

the

me

as

ure

s o

f a

ng

le

3 a

nd

4 i

s e

qu

al

to t

he

me

as

ure

of

an

gle

1;

m∠

3 +

m∠

4 =

97

001_

018_

GE

OC

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C04

_890

513.

indd

154/

12/0

812

:35:

03A

M




An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

16

G

lenc

oe G

eom

etry

Enri

chm

ent

Fin

din

g A

ng

le M

easu

res i

n T

rian

gle

sY

ou c

an u

se a

lgeb

ra t

o so

lve

prob

lem

s in

volv

ing

tria

ngl

es.

In

tri

angl

e A

BC

, m∠

A i

s tw

ice

m∠

B, a

nd

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

= 1

80

2x +

x +

(x +

8) =

180

4x

+ 8

= 1

80

4x =

172

x =

43

So,

m∠

A =

2(4

3) o

r 86

, m∠

B =

43,

an

d m

∠C

= 4

3 +

8 o

r 51

.

Sol

ve e

ach

pro

ble

m.

1. I

n t

rian

gle

DE

F, m

∠E

is

thre

e ti

mes

2.

In

tri

angl

e R

ST

, m∠

T i

s 5

mor

e th

an

m∠

D, a

nd

m∠

F i

s 9

less

th

an m

∠E

. m

∠R

, an

d m

∠S

is

10 l

ess

than

m∠

T.

Wh

at i

s th

e m

easu

re o

f ea

ch a

ngl

e?

Wh

at i

s th

e m

easu

re o

f ea

ch a

ngl

e?

3. I

n t

rian

gle

JK

L, m

∠K

is

fou

r ti

mes

4.

In

tri

angl

e X

YZ

, m∠

Z i

s 2

mor

e th

an t

wic

e m

∠J

, an

d m

∠L

is

five

tim

es m

∠J

. m

∠X

, an

d m

∠Y

is

7 le

ss t

han

tw

ice

m∠

X.

Wh

at i

s th

e m

easu

re o

f ea

ch a

ngl

e?

Wh

at i

s th

e m

easu

re o

f ea

ch a

ngl

e?

5. I

n t

rian

gle

GH

I, m

∠H

is

20 m

ore

than

6.

In

tri

angl

e M

NO

, m∠

M i

s eq

ual

to

m∠

N,

m∠

G, a

nd

m∠

G i

s 8

mor

e th

an m

∠I.

an

d m

∠O

is

5 m

ore

than

th

ree

tim

es

Wh

at i

s th

e m

easu

re o

f ea

ch a

ngl

e?

m

∠N

. Wh

at i

s th

e m

easu

re o

f ea

ch a

ngl

e?

7. I

n t

rian

gle

ST

U, m

∠U

is

hal

f m

∠T

, 8.

In

tri

angl

e P

QR

, m∠

P i

s eq

ual

to

and

m∠

S i

s 30

mor

e th

an m

∠T

. Wh

at

m∠

Q, a

nd

m∠

R i

s 24

les

s th

an m

∠P

.is

th

e m

easu

re o

f ea

ch a

ngl

e?

Wh

at i

s th

e m

easu

re o

f ea

ch a

ngl

e?

9. W

rite

you

r ow

n p

robl

ems

abou

t m

easu

res

of t

rian

gles

.

4-2

Exam

ple

m∠

D =

27,

m∠

E =

81,

m∠

F =

72

m∠

R =

60,

m∠

S =

55,

m∠

T =

65

m∠

J =

18,

m∠

K =

72,

m∠

L =

90

m∠

X =

37,

m∠

Y =

67,

m∠

Z =

76

m∠

G =

56,

m∠

H =

76,

m∠

I =

48

m∠

M =

m∠

N =

35,

m∠

O =

110

m∠

S =

90,

m∠

T =

60,

m∠

U =

30

m∠

P =

m∠

Q =

68,

m∠

R =

44

See s

tud

en

ts’

wo

rk.

001_

018_

GE

OC

RM

C04

_890

513.

indd

164/

12/0

812

:35:

09A

M

Lesson 4-2


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

17

G

lenc

oe G

eom

etry

Gra

phin

g Ca

lcul

ator

Act

ivit

yC

ab

ri J

un

ior:

An

gle

s o

f Tri

an

gle

s

Cab

ri J

un

ior

can

be

use

d to

in

vest

igat

e th

e re

lati

onsh

ips

betw

een

th

e an

gles

in

a t

rian

gle.

U

se C

abri

Ju

nio

r to

in

vest

igat

e th

e su

m o

f th

e m

easu

res

of t

he

angl

es o

f a

tria

ngl

e an

d t

o in

vest

igat

e th

e m

easu

re o

f th

e ex

teri

or a

ngl

e in

a

tria

ngl

e in

rel

atio

n t

o it

s re

mot

e in

teri

or a

ngl

es.

Ste

p 1

U

se C

abri

Ju

nio

r to

dra

w a

tri

angl

e.

• S

elec

t F

2 T

rian

gle

to d

raw

a t

rian

gle.

•

Mov

e th

e cu

rsor

to

wh

ere

you

wan

t th

e fi

rst

vert

ex. T

hen

pre

ss E

NT

ER

.

• R

epea

t th

is p

roce

dure

to

dete

rmin

e th

e n

ext

two

vert

ices

of

the

tria

ngl

e.

Ste

p 2

L

abel

eac

h v

erte

x of

th

e tr

ian

gle.

•

Sel

ect

F5

Alp

h-n

um

.

• M

ove

the

curs

or t

o a

vert

ex a

nd

pres

s E

NT

ER

. En

ter

A a

nd

pres

s E

NT

ER

aga

in.

•

Rep

eat

this

pro

cedu

re t

o la

bel

vert

ex B

an

d ve

rtex

C.

Ste

p 3

D

raw

an

ext

erio

r an

gle

of �

AB

C.

•

Sel

ect

F2

Lin

e to

dra

w a

lin

e th

rou

gh −−

− B

C .

•

Sel

ect

F2

Poi

nt,

Poi

nt

on t

o dr

aw a

poi

nt

on �

��

BC

so

that

C i

s be

twee

n B

an

d th

e n

ew p

oin

t.

• S

elec

t F

5 A

lph

-nu

m t

o la

bel

the

poin

t D

.

Ste

p 4

F

ind

the

mea

sure

of

the

thre

e in

teri

or a

ngl

es o

f �

AB

C a

nd

the

exte

rior

an

gle,

∠A

CD

.

• S

elec

t F

5 M

easu

re, A

ngl

e.

• T

o fi

nd

the

mea

sure

of

∠A

BC

, sel

ect

poin

ts A

, B, a

nd

C

(wit

h t

he

vert

ex B

as

the

seco

nd

poin

t se

lect

ed).

•

Use

th

is m

eth

od t

o fi

nd

the

rem

ain

ing

angl

e m

easu

res.

Exer

cise

s

An

alyz

e yo

ur

dra

win

g.

1. U

se F

5 C

alcu

late

to

fin

d th

e su

m o

f th

e m

easu

res

of t

he

thre

e in

teri

or a

ngl

es.

2. M

ake

a co

nje

ctu

re a

bou

t th

e su

m o

f th

e in

teri

or a

ngl

es o

f a

tria

ngl

e.

3. C

han

ge t

he

dim

ensi

ons

of t

he

tria

ngl

e by

mov

ing

poin

t A

. (P

ress

CLE

AR

so

the

poin

ter

beco

mes

a b

lack

arr

ow. M

ove

the

poin

ter

clos

e to

poi

nt

A u

nti

l th

e ar

row

bec

omes

tr

ansp

aren

t an

d po

int

A i

s bl

inki

ng.

Pre

ss A

LP

HA

to

chan

ge t

he

arro

w t

o a

han

d. T

hen

m

ove

the

poin

t.)

Is y

our

con

ject

ure

sti

ll t

rue?

4. W

hat

is

the

mea

sure

of

∠A

CD

? W

hat

is

the

sum

of

the

mea

sure

s of

th

e tw

o re

mot

e in

teri

or a

ngl

es (

∠A

BC

an

d ∠

BA

C)?

5. M

ake

a co

nje

ctu

re a

bou

t th

e re

lati

onsh

ip b

etw

een

th

e m

easu

re o

f an

ext

erio

r an

gle

of a

tr

ian

gle

and

its

two

rem

ote

inte

rior

an

gles

.

6. C

han

ge t

he

dim

ensi

ons

of t

he

tria

ngl

e by

mov

ing

poin

t A

. Is

you

r co

nje

ctu

re s

till

tru

e?

4-2

Exam

ple

yes

Th

e s

um

of

the m

easu

res o

f th

e i

nte

rio

r an

gle

s o

f a t

rian

gle

is 1

80

°

yes

See s

tud

en

ts’ w

ork

Th

e m

easu

re o

f an

exte

rio

r an

gle

is

eq

ual to

th

e s

um

of

the m

easu

res o

f it

s t

wo

rem

ote

in

teri

or

an

gle

s.

180

°

001_

018_

GE

OC

RM

C04

_890

513.

indd

175/

22/0

88:

04:1

4P

M




Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

18

G

lenc

oe G

eom

etry

Geo

met

er’s

Ske

tchp

ad A

ctiv

ity

An

gle

s o

f Tri

an

gle

s

Th

e G

eom

eter

’s S

ketc

hpa

d ca

n b

e u

sed

to i

nve

stig

ate

the

rela

tion

ship

s be

twee

n t

he

angl

es

in a

tri

angl

e.

U

se T

he

Geo

met

er’s

Sk

etch

pad

to

inve

stig

ate

the

sum

of

the

mea

sure

s of

th

e an

gles

of

a tr

ian

gle

and

to

inve

stig

ate

the

mea

sure

of

the

exte

rior

an

gle

in a

tri

angl

e in

rel

atio

n t

o it

s re

mot

e in

teri

or a

ngl

es.

Ste

p 1

U

se T

he

Geo

met

er’s

Ske

tch

pad

to d

raw

a t

rian

gle.

• C

onst

ruct

a r

ay b

y se

lect

ing

the

Ray

too

l fr

om t

he

tool

bar.

Fir

st, c

lick

wh

ere

you

wan

t th

e fi

rst

poin

t. T

hen

cli

ck o

n a

sec

ond

poin

t to

dra

w t

he

ray.

•

Nex

t, s

elec

t th

e S

egm

ent

tool

fro

m t

he

tool

bar.

Use

th

e en

dpoi

nt

of t

he

ray

as

the

firs

t po

int

for

the

segm

ent

and

clic

k on

a s

econ

d po

int

to c

onst

ruct

th

e se

gmen

t.

• C

onst

ruct

an

oth

er s

egm

ent

join

ing

the

seco

nd

poin

t of

th

e n

ew s

egm

ent

to a

po

int

on t

he

ray.

Ste

p 2

D

ispl

ay t

he

labe

ls f

or e

ach

poi

nt.

U

se t

he

poin

ter

tool

to

sele

ct a

ll

fou

r po

ints

. Dis

play

th

e la

bels

by

sel

ecti

ng

Sh

ow L

abel

s fr

om

the

Dis

pla

y m

enu

.

Ste

p 3

F

ind

the

mea

sure

s of

th

e th

ree

inte

rior

an

gles

an

d th

e on

e ex

teri

or

angl

e. T

o fi

nd

the

mea

sure

of

angl

e A

BC

, use

th

e S

elec

tion

Arr

ow t

ool

to s

elec

t po

ints

A, B

, an

d C

(w

ith

th

e ve

rtex

B a

s th

e se

con

d po

int

sele

cted

). T

hen

, un

der

the

Mea

sure

m

enu

, sel

ect

An

gle.

Use

th

is m

eth

od

to f

ind

the

rem

ain

ing

angl

e m

easu

res.

Exer

cise

sA

nal

yze

you

r d

raw

ing.

1. W

hat

is

the

sum

of

the

mea

sure

s of

th

e th

ree

inte

rior

an

gles

?

2. M

ake

a co

nje

ctu

re a

bou

t th

e su

m o

f th

e in

teri

or a

ngl

es o

f a

tria

ngl

e.

3. C

han

ge t

he

dim

ensi

ons

of t

he

tria

ngl

e by

sel

ecti

ng

poin

t A

wit

h t

he

poin

ter

tool

an

d m

ovin

g it

. Is

you

r co

nje

ctu

re s

till

tru

e?

4. W

hat

is

the

mea

sure

of ∠

AC

D?

Wh

at i

s th

e su

m o

f th

e m

easu

res

of t

he

two

rem

ote

inte

rior

an

gles

(∠

AB

C a

nd

∠B

AC

)? 5

. Mak

e a

con

ject

ure

abo

ut

the

rela

tion

ship

bet

wee

n t

he

mea

sure

of

an

exte

rior

an

gle

of a

tri

angl

e an

d it

s tw

o re

mot

e in

teri

or a

ngl

es.

6. C

han

ge t

he

dim

ensi

ons

of t

he

tria

ngl

e by

sel

ecti

ng

poin

t A

wit

h t

he

poin

ter

tool

an

d m

ovin

g it

. Is

you

r co

nje

ctu

re s

till

tru

e?

m/

AB

C =

60

.00 °

m/

BA

C =

58

.19 °

m/

AC

B =

61

.81 °

m/

AC

D =

11

8.19

°m

/A

BC

+ m

/B

AC

+ m

/A

CB

= 18

0.00

°m

/A

BC

+ m

/B

AC

= 11

8.19

°

BC

D

A

4-2

Exam

ple

180

°

yes

See s

tud

en

ts’

wo

rk.

Th

e s

um

of

the m

easu

res o

f th

e i

nte

rio

r an

gle

s o

f a t

rian

gle

is 1

80

°.

Th

e m

easu

res o

f an

exte

rio

r an

gle

is e

qu

al to

th

e s

um

of

the m

easu

re o

f it

s t

wo

rem

ote

in

teri

or

an

gle

s.

yes

001_

018_

GE

OC

RM

C04

_890

513.

indd

185/

22/0

88:

04:1

7P

M


NA

ME

DA

TE

PE

RIO

D

Lesson 4-3

Cha

pte

r 4

19

G

lenc

oe G

eom

etry

Co

ng

ruen

ce a

nd

Co

rres

po

nd

ing

Par

ts

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

th

e fi

gure

, �A

BC

� �

RS

T.

If

�X

YZ

� �

RS

T, n

ame

the

pai

rs o

f co

ngr

uen

t an

gles

an

d c

ongr

uen

t si

des

.

∠X

� ∠

R, ∠

Y �

∠S

, ∠Z

� ∠

T −

−

XY

� −

−

RS

, −−

XZ

� −

−

RT

, −−

YZ

� −

−

ST

Exer

cise

sS

how

th

at t

he

pol

ygon

s ar

e co

ngr

uen

t b

y id

enti

fyin

g al

l co

ngr

uen

t co

rres

pon

din

g p

arts

. Th

en w

rite

a c

ongr

uen

ce s

tate

men

t.

1.

CA

B

LJK

2.

C

D

A

B

3. K J

L M

4. F

GL

K JE

5.

BD

CA

6.

R TUS

�A

BC

� �

DE

F.

7. F

ind

the

valu

e of

x.

8. F

ind

the

valu

e of

y.

Y

XZ

T

SR

AC

B

R

TS

4-3

Stud

y G

uide

and

Inte

rven

tion

Co

ng

ruen

t Tri

an

gle

s

Th

ird

An

gle

s

Th

eo

rem

If t

wo

an

gle

s o

f o

ne

tria

ng

le a

re c

on

gru

en

t to

tw

o a

ng

les

of

a s

eco

nd

tria

ng

le,

the

n t

he

th

ird

an

gle

s o

f th

e t

ria

ng

les a

re c

on

gru

en

t.

Exam

ple

2x+

y(2

y-

5)°

75°

65°

40°

96.6

90.6

64.3

∠A

� ∠

J;

∠B

� ∠

K;

∠A

� ∠

D;

∠A

BC

� ∠

BC

D

∠

J �

∠L

; ∠

JK

M �

∠LM

K;

∠C

� ∠

L; −

−

A

B �

−−

JK

;

∠

AC

B �

∠C

BD

; −

−

A

C �

−

−

B

D

∠

KM

J �

∠M

KL; −

−

K

J �

−

−

LM

−

−

B

C �

−−

K

L ; −

−

A

C �

−−

JL

−

−

A

B �

−

−

C

D

−

−

K

L �

−

−

JM

�A

BC

� �

JK

L

�

AB

C �

�D

CB

�JK

M �

�LM

K

27.8

35

∠E

� ∠

J;

∠F

� ∠

K;

∠A

� ∠

D;

∠R

� ∠

T;

∠G

� ∠

L; −

−

E

F �

−−

JK

;

∠

AB

C �

∠D

CB

;

∠

RS

U �

∠T

SU

;

−

−

E

G �

−−

JL ; −

−

FG

� −

−

K

L

∠

AC

B �

∠D

BC

;

∠

RU

S �

∠T

US

;

�FG

E �

�K

LJ

−

−

A

B �

−−

D

C ; −

−

A

C �

−−

D

B ;

−

−

R

U �

−−

TU

; −

−

R

S �

−−

TS ;

−

−

B

C �

−−

C

B ;

�A

BC

� �

DC

B

−

−

S

U �

−−

S

U ;

�R

SU

� �

TS

U

019_

036_

GE

OC

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C04

_890

513.

indd

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12/0

812

:31:

15A

M




An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

20

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Co

ng

ruen

t Tri

an

gle

s

4-3

Exam

ple

Pro

ve T

rian

gle

s C

on

gru

ent

Tw

o tr

ian

gles

are

con

gru

ent

if a

nd

only

if

thei

r co

rres

pon

din

g pa

rts

are

con

gru

ent.

Cor

resp

ondi

ng

part

s in

clu

de c

orre

spon

din

g an

gles

an

d co

rres

pon

din

g si

des.

Th

e ph

rase

“if

an

d on

ly i

f” m

ean

s th

at b

oth

th

e co

ndi

tion

al a

nd

its

con

vers

e ar

e tr

ue.

For

tri

angl

es, w

e sa

y, “

Cor

resp

ondi

ng

part

s of

con

gru

ent

tria

ngl

es a

re

con

gru

ent,

” or

CP

CT

C.

W

rite

a t

wo-

colu

mn

pro

of.

Giv

en:

−−

AB

� −−

− C

B ,

−−−

AD

� −−

− C

D , ∠

BA

D �

∠B

CD

−−

− B

D b

isec

ts ∠

AB

C P

rove

: �A

BD

� �

CB

D

Pro

of:

Sta

tem

ent

Rea

son

1. −

−

AB

� −−

− C

B ,

−−−

AD

� −−

− C

D

2. −−

− B

D �

−−−

BD

3.

∠B

AD

� ∠

BC

D4.

∠A

BD

� ∠

CB

D5.

∠B

DA

� ∠

BD

C6.

�A

BD

� �

CB

D

1. G

iven

2. R

efle

xive

Pro

pert

y of

con

gru

ence

3. G

iven

4. D

efin

itio

n o

f an

gle

bise

ctor

5. T

hir

d A

ngl

es T

heo

rem

6. C

PC

TC

Exer

cise

sW

rite

a t

wo-

colu

mn

pro

of.

1. G

iven

: ∠A

� ∠

C, ∠

D �

∠B

, −−

− A

D �

−−−

CB

, −

−

AE

� −−

− C

E ,

−−

AC

bis

ects

−−−

BD

P

rove

: �A

ED

� �

CE

BP

roo

f:

Sta

tem

en

tsR

easo

ns

1.

∠A

� ∠

C,

∠D

� ∠

B

2.

∠A

ED

� ∠

CE

B

3.

−−

A

D �

−−

C

B ,

−−

A

E �

−−

C

E

4.

−−

D

E �

−

−

B

E

5.

�A

ED

� �

CE

B

1.

Giv

en

2.

Vert

ical

an

gle

s �

3.

Giv

en

4.

Defi

nit

ion

of

seg

men

t b

isecto

r

5.

CP

CT

C

Wri

te a

par

agra

ph

pro

of.

2. G

iven

: −−

− B

D b

isec

ts ∠

AB

C a

nd

∠A

DC

, −

−

AB

� −−

− C

B ,

−−

AB

� −−

− A

D ,

−−−

CB

� −−

− D

C

Pro

ve: �

AB

D �

�C

BD

We a

re g

iven

−−

B

D b

isects

∠A

BC

an

d ∠

AD

C.

Th

ere

fore

∠

AB

D �

∠C

BD

an

d ∠

AD

B �

∠C

DB

by t

he d

efi

nit

ion

o

f an

gle

bis

ecto

rs.

By t

he T

hir

d A

ng

le T

heo

rem

, w

e

fin

d t

hat

∠A

� ∠

C.

We a

re g

iven

th

at

−−

A

B �

−−

C

B ,

−−

A

B �

−

−

A

D ,

an

d −

−

C

B �

−−

D

C .

Usin

g t

he s

ub

sti

tuti

on

pro

pert

y,

we c

an

dete

rmin

e t

hat

−−

A

D �

−−

C

D .

Fin

ally,

−−

B

D �

−

−

B

D u

sin

g t

he R

efl

exiv

e P

rop

ert

y o

f co

ng

ruen

ce.

Th

ere

fore

�A

BD

� �

CB

D b

y C

PC

TC

.

019_

036_

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OC

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C04

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513.

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12/0

812

:31:

25A

M


NA

ME

DA

TE

PE

RIO

D

Lesson 4-3

Cha

pte

r 4

21

G

lenc

oe G

eom

etry

4-3

Sh

ow t

hat

pol

ygon

s ar

e co

ngr

uen

t b

y id

enti

fyin

g al

l co

ngr

uen

t co

rres

pon

din

g p

arts

. Th

en w

rite

a c

ongr

uen

ce s

tate

men

t.

1.

L

P

J

S

V

T

2.

In

th

e fig

ur

e, �

AB

C �

�F

DE

. 3

. Fin

d th

e va

lue

of x

. 36

4. F

ind

the

valu

e of

y.

48

5. P

RO

OF

Wri

te a

tw

o-co

lum

n p

roof

.

G

iven

: −

−−

AB

� −

−−

CB

, −−

−

AD

� −

−−

CD

, ∠A

BD

� ∠

CB

D,

∠A

DB

� ∠

CD

B

Pro

ve: �

AB

D �

�C

BD

Pro

of:

Sta

tem

en

ts

Reaso

ns

1.

−−

A

B �

−−

C

B ,

−−

A

D �

−

−

C

D

1.

Giv

en

2.

∠A

BD

� ∠

CB

D,

∠A

DB

� ∠

CD

B

2.

Giv

en

3.

∠A

� ∠

C

3.

Th

ird

An

gle

Th

eo

rem

4.

�A

BD

� �

CB

D

4.

CP

CT

C

Skill

s Pr

acti

ceC

on

gru

en

t Tri

an

gle

s

D

E G

F

108°

48°

y°

(2x

-y)

°

AF

BC

ED

∠J �

∠T;

∠P

� ∠

V;

∠E

DF �

∠FD

G;

∠E

� ∠

G;

∠L �

∠S

; −

−

JP

� −

−

TV

∠

EFD

� ∠

GFD

; −

−

D

E �

−

−

D

G ;

−−

P

L �

−−

V

S ;

−−

JL �

−−

TS

−

−

G

F �

−−

E

F ;

�D

EF �

�D

GF

�JP

L �

�TV

S

019_

036_

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29/0

811

:59:

24A

M




Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

22

G

lenc

oe G

eom

etry

Prac

tice

Co

ng

ruen

t Tri

an

gle

s

Sh

ow t

hat

th

e p

olyg

ons

are

con

gru

ent

by

ind

enti

fyin

g al

l co

ngr

uen

t co

rres

pon

din

g p

arts

. Th

en w

rite

a c

ongr

uen

ce s

tate

men

t.

1.

D

R

SC

A

B

2.

MN

L

P

Q

Pol

ygon

AB

CD

� p

olyg

on P

QR

S.

3. F

ind

the

valu

e of

x.

4. F

ind

the

valu

e of

y.

5. P

RO

OF

Wri

te a

tw

o-co

lum

n p

roof

.G

iven

: ∠P

� ∠

R, ∠

PS

Q �

∠R

SQ

, −−

− P

Q �

−−−

RQ

, −

−

PS

� −

−

RS

Pro

ve: �

PQ

S �

�R

QS

Pro

of:

Sta

tem

en

ts

Reaso

ns

1.

∠P

� ∠

R,

∠P

SQ

� ∠

RS

Q

1.

Giv

en

2.

∠P

QS

� ∠

RQ

S

2.

Th

ird

An

gle

Th

eo

rem

3.

−−

P

Q �

−−

R

Q ,

−−

P

S �

−−

R

S

3.

Giv

en

4.

−−

Q

S �

−−

Q

S

4.

Refl

exiv

e P

rop

ert

y

5.

�P

QS

� �

RQ

S

5.

CP

CT

C

6. Q

UIL

TIN

G

a. I

ndi

cate

th

e tr

ian

gles

th

at a

ppea

r to

be

con

gru

ent.

b.

Nam

e th

e co

ngr

uen

t an

gles

an

d co

ngr

uen

t si

des

of a

pai

r of

co

ngr

uen

t tr

ian

gles

.

B

A I

E FH

G

CD

4-3

(2x

+4)

° (3y

-3)

80°

100°

1012

∠A

� ∠

D;

∠B

� ∠

R

∠L �

∠Q

; ∠

M �

∠P

∠C

� ∠

S;

−−

A

B �

−−

R

D

∠M

NL �

∠P

NQ

;

−−

B

C �

−−

R

S ;

−−

A

C �

−−

S

D

−−

LM

� −

−

P

Q ;

−−

M

N �

−−

N

P

�A

BC

� �

DR

S

−−

LM

� −

−

N

Q

�LM

N �

�Q

PN

�A

BI

� �

EB

F,

�C

BD

� �

HB

G

S

am

ple

an

sw

er:

∠A

� ∠

E,

∠A

BI

� ∠

EB

F,

∠I

� ∠

F;

−−

A

B �

−−

E

B ,

−−

B

I �

−−

B

F ,

−−

A

I �

−−

E

F

x =

48

y =

5

019_

036_

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OC

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C04

_890

513.

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12/0

812

:31:

41A

M


NA

ME

DA

TE

PE

RIO

D

Lesson 4-3

Cha

pte

r 4

23

G

lenc

oe G

eom

etry

Wor

d Pr

oble

m P

ract

ice

Co

ng

ruen

t Tri

an

gle

s

1. P

ICTU

RE

HA

NG

ING

Can

dice

hu

ng

a pi

ctu

re t

hat

was

in

a t

rian

gula

r fr

ame

on h

er b

edro

om w

all.

Befo

reAB

C

A

B

CAfte

r

O

ne

day,

it

fell

to

the

floo

r, b

ut

did

not

br

eak

or b

end.

Th

e fi

gure

sh

ows

the

obje

ct b

efor

e an

d af

ter

the

fall

. Lab

el

the

vert

ices

on

th

e fr

ame

afte

r th

e fa

ll

acco

rdin

g to

th

e “b

efor

e” f

ram

e.

2. S

IER

PIN

SKI’S

TR

IAN

GLE

Th

e fi

gure

be

low

is

a po

rtio

n o

f S

ierp

insk

i’s

Tri

angl

e. T

he

tria

ngl

e h

as t

he

prop

erty

th

at a

ny

tria

ngl

e m

ade

from

an

y co

mbi

nat

ion

of

edge

s is

equ

ilat

eral

. H

ow m

any

tria

ngl

es i

n t

his

por

tion

are

co

ngr

uen

t to

th

e bl

ack

tria

ngl

e at

th

e bo

ttom

cor

ner

?

3. Q

UIL

TIN

G S

tefa

n d

rew

th

is p

atte

rn f

or

a pi

ece

of h

is q

uil

t. I

t is

mad

e u

p of

co

ngr

uen

t is

osce

les

righ

t tr

ian

gles

. He

drew

on

e tr

ian

gle

and

then

rep

eate

dly

drew

it

all

the

way

aro

un

d.

45˚

W

hat

are

th

e m

issi

ng

mea

sure

s of

th

e an

gles

of

the

tria

ngl

e?

4. M

OD

ELS

Dan

a bo

ugh

t a

mod

el a

irpl

ane

kit.

Wh

en h

e op

ened

th

e bo

x, t

hes

e tw

o co

ngr

uen

t tr

ian

gula

r pi

eces

of

woo

d fe

ll

out

of i

t.

A

B

CQ

R

P

I

den

tify

th

e tr

ian

gle

that

is

con

gru

ent

to

�A

BC

.

5. G

EOG

RA

PHY

Igo

r n

otic

ed o

n a

map

th

at t

he

tria

ngl

e w

hos

e ve

rtic

es a

re t

he

supe

rmar

ket,

th

e li

brar

y, a

nd

the

post

of

fice

(�

SL

P)

is c

ongr

uen

t to

th

e tr

ian

gle

wh

ose

vert

ices

are

Igo

r’s

hom

e,

Jaco

b’s

hom

e, a

nd

Ben

’s h

ome

(�IJ

B).

T

hat

is,

�S

LP

� �

IJB

.

a. T

he

dist

ance

bet

wee

n t

he

supe

rmar

ket

and

the

post

off

ice

is

1 m

ile.

Wh

ich

pat

h a

lon

g th

e tr

ian

gle

�IJ

B i

s co

ngr

uen

t to

th

is?

b

. T

he

mea

sure

of

∠L

PS

is

40. I

den

tify

th

e an

gle

that

is

con

gru

ent

to t

his

an

gle

in �

IJB

.

4-3 11

45 a

nd

90

�Q

RP

the p

ath

betw

een

Ig

or’

s

ho

me a

nd

Ben

’s h

om

e

∠JB

I

019_

036_

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C04

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513.

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234/

12/0

812

:31:

51A

M




An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

24

G

lenc

oe G

eom

etry

Enri

chm

ent

4-3

Overl

ap

pin

g T

rian

gle

sS

omet

imes

wh

en t

rian

gles

ove

rlap

eac

h o

ther

, th

ey s

har

e co

rres

pon

din

g pa

rts.

Wh

en t

ryin

g to

pic

ture

th

e co

rres

pon

din

g pa

rts,

try

red

raw

ing

the

tria

ngl

es s

epar

atel

y.

In t

he

figu

re �

AB

D �

�D

CA

an

d m

∠B

DA

= 3

0°.

Fin

d m

∠C

AD

an

d m

∠C

DA

.

30°

Fir

st, r

edra

w �

AB

D a

nd

�D

CA

lik

e th

is.

30°

Now

you

can

see

th

at ∠

BD

A �

∠C

AD

bec

ause

th

ey a

re c

orre

spon

din

g pa

rts

of c

ongr

uen

t tr

ian

gles

, so

m∠

CA

D =

30.

Usi

ng

the

Tri

angl

e S

um

Th

eore

m, 1

80 -

(90

+ 3

0) =

60,

so

m∠

CD

A =

60.

Exer

cise

s 1

. In

th

e fi

gure

�A

BC

� �

CD

B, m

∠A

BC

= 1

10, a

nd

m∠

D =

20.

Fin

d m

∠A

CB

an

d m

∠D

CB

.

2. W

rite

a t

wo

colu

mn

pro

of.

Giv

en: ∠

A a

nd

∠B

are

rig

ht

angl

es, ∠

AC

D �

∠B

DC

, −−

− A

D �

−−

− B

C ,

−−

AC

� −−

− B

D

Pro

ve: �

AD

C �

�C

BD

Pro

of:

Sta

tem

en

ts

Reaso

ns

1.

∠A

an

d ∠

B a

re r

igh

t an

gle

s

1.

Giv

en

2.

∠A

� ∠

B

2.

All r

t �

are

�

3.

∠A

CD

� ∠

BD

C

3.

Giv

en

4.

∠A

DC

� ∠

BC

D

4.

Th

ird

An

gle

Th

eo

rem

5.

−−

A

D �

−−

B

C ,

−−

A

C �

−−

B

D

5.

Giv

en

6.

−−

D

C �

−−

C

D

6.

Refl

exiv

e p

rop

ert

y

7.

�A

DC

� �

CB

D

7.

CP

CT

C

Exam

ple

m∠

AC

B =

m∠

DB

C =

50

019_

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NA

ME

DA

TE

PE

RIO

D

Lesson 4-4

Cha

pte

r 4

25

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

Pro

vin

g T

rian

gle

s C

on

gru

en

t—S

SS

, S

AS

SSS

Post

ula

te Y

ou k

now

th

at t

wo

tria

ngl

es a

re c

ongr

uen

t if

cor

resp

ondi

ng

side

s ar

e co

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

e ar

e co

ngr

uen

t to

th

e si

des

of t

he

seco

nd

tria

ngl

e.

W

rite

a t

wo-

colu

mn

pro

of.

Giv

en:

−−

AB

� −−

− D

B a

nd

C i

s th

e m

idpo

int

of −−

− A

D .

Pro

ve: �

AB

C �

�D

BC

B C

DA

Sta

tem

ents

R

easo

ns

1. −

−

AB

� −−

− D

B

1. G

iven

2. C

is

the

mid

poin

t of

−−−

AD

. 2.

Giv

en3.

−−

AC

� −−

− D

C

3. M

idpo

int

Th

eore

m4.

−−−

BC

� −−

− B

C

4. R

efle

xive

Pro

pert

y of

�5.

�A

BC

� �

DB

C

5. S

SS

Pos

tula

te

Exer

cise

s

Wri

te a

tw

o-co

lum

n p

roof

.

1.

BY

CA

XZ

Giv

en:

AB

�−

−X

Y ,

−−

AC

�−

−X

Z ,

−−−

BC

�−

−Y

Z

Pro

ve:�

AB

C�

�X

YZ

2.

TU

RS

Giv

en:

−−

RS

� −−

− U

T ,

−−

RT

� −−

− U

S

Pro

ve:

�R

ST

� �

UT

S

4-4

SS

S P

os

tula

teIf

th

ree

sid

es o

f o

ne

tria

ng

le a

re c

on

gru

en

t to

th

ree

sid

es o

f a

se

co

nd

tria

ng

le,

the

n t

he

tria

ng

les a

re c

on

gru

en

t.

Exam

ple

Sta

tem

en

ts

Reaso

ns

Sta

tem

en

ts

Reaso

ns

1.

−−

A

B �

−−

X

Y

1.

Giv

en

−

−

A

C �

−−

X

Z

−

−

B

C �

−−

Y

Z

2.

�A

BC

� �

XY

Z

2.

SS

S P

ost.

1.

−−

R

S �

−−

U

T

1.

Giv

en

−

−

R

T �

−−

U

S

2.

−−

S

T �

−−

TS

2

. R

efl

. P

rop

.

3.

�R

ST �

�U

TS

3.

SS

S P

ost.

019_

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Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

26

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Pro

vin

g T

rian

gle

s C

on

gru

en

t—S

SS

, S

AS

SAS

Post

ula

te A

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

.

F

or e

ach

dia

gram

, det

erm

ine

wh

ich

pai

rs o

f tr

ian

gles

can

be

pro

ved

con

gru

ent

by

the

SA

S P

ostu

late

.

a.

A BC

X YZ

b

. D

H

FE

G

J

c.

P

Q

S

1

2R

I

n �

AB

C, t

he

angl

e is

not

T

he

righ

t an

gles

are

T

he

incl

ude

d an

gles

, “i

ncl

ude

d” b

y th

e si

des

−−

AB

c

ongr

uen

t an

d th

ey a

re t

he

∠1

and

∠2,

are

con

gru

ent

and

−−

AC

. So

the

tria

ngl

es

in

clu

ded

angl

es f

or t

he

b

ecau

se t

hey

are

ca

nn

ot b

e pr

oved

con

gru

ent

c

ongr

uen

t si

des.

a

lter

nat

e in

teri

or a

ngl

es

by t

he

SA

S P

ostu

late

. �

DE

F �

�J

GH

by

the

f

or t

wo

para

llel

lin

es.

S

AS

Pos

tula

te.

�P

SR

� �

RQ

P b

y th

e

S

AS

Pos

tula

te.

Exer

cise

s

4-4

SA

S P

os

tula

te

If t

wo

sid

es a

nd

th

e in

clu

de

d a

ng

le o

f o

ne

tria

ng

le a

re c

on

gru

en

t to

tw

o

sid

es a

nd

th

e in

clu

de

d a

ng

le o

f a

no

the

r tr

ian

gle

, th

en

th

e t

ria

ng

les a

re

co

ng

rue

nt.

Exam

ple

Wri

te t

he

spec

ifie

d t

ype

of p

roof

. 1

. Wri

te a

tw

o co

lum

n p

roof

. 2

. Wri

te a

tw

o-co

lum

n p

roof

.

G

iven

: NP

= P

M,

−−−

NP

⊥ −

−

PL

G

iven

: AB

= C

D,

−−

AB

‖ −−

− C

D

Pro

ve: �

NP

L �

�M

PL

P

rove

: �A

CD

� �

CA

B

Pro

of:

Pro

of:

3. W

rite

a p

arag

raph

pro

of.

Giv

en: V

is

the

mid

poin

t of

−−

YZ

.

V i

s th

e m

idpo

int

of −−

− W

X .

Pro

ve: �

XV

Z �

�W

VY

S

ince V

is t

he m

idp

oin

t o

f −

−

Y

Z a

nd

th

e m

idp

oin

t o

f −

−

W

X ,

by t

he d

efi

nit

ion

of

mid

po

int,

Y

V =

VZ

an

d W

V =

VX

. S

ince ∠

YV

W a

nd

∠X

VZ

are

vert

ical

an

gle

s,

by t

he V

ert

ical

An

gle

Th

eo

rem

th

e a

ng

les a

re c

on

gru

en

t. T

here

fore

, b

y S

AS

, �

XV

Z �

�W

VY

Sta

tem

en

tsR

easo

ns

1.

NP

= P

M, −

−

N

P ⊥

−−

P

L

2.

∠M

PL i

s a

rig

ht

an

gle

∠N

PL i

s a

rig

ht

an

gle

3.

∠M

PL

� ∠

NP

L

4.

PL =

PL

5.

�N

PL �

�M

PL

1.

Giv

en

2.

perp

en

dic

ula

r lin

es

form

rig

ht

an

gle

s

3.

all r

igh

t an

gle

s a

re

co

ng

ruen

t

4.

Refl

exiv

e P

rop

ert

y

of

co

ng

ruen

ce

5.

SA

S

Sta

tem

en

tsR

easo

ns

1.

AB

= C

D, −

−

A

B ‖

−−

C

D

2.

∠B

AC

� ∠

DC

A

3.

AC

= A

C

4.

�A

CD

� �

CA

B

1.

Giv

en

2.

Alt

ern

ate

In

teri

or

An

gle

s T

heo

rem

3.

Re

fle

xiv

e P

rop

ert

y

of

co

ng

rue

nc

e

4.

SA

S

019_

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NA

ME

DA

TE

PE

RIO

D

Lesson 4-4

Cha

pte

r 4

27

G

lenc

oe G

eom

etry

Det

erm

ine

wh

eth

er �

AB

C �

�K

LM

. Exp

lain

.

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)

PRO

OF

Wri

te t

he

spec

ifie

d t

ype

of p

roof

.

3. W

rite

a f

low

pro

of.

G

iven

: −

−

PR

� −−

− D

E ,

−−

PT

� −−

− D

F

∠R

� ∠

E, ∠

T �

∠F

P

rove

: �

PR

T �

�D

EF

PR�

DE

Give

n

PT�

DF

Give

n

∠R

�∠

EGi

ven

∠P

�∠

DTh

ird A

ngle

Theo

rem

�PR

T�

�D

EFSA

S

∠T

�∠

FGi

ven

4. W

rite

a t

wo-

colu

mn

pro

of.

Giv

en:

−−

AB

� −−

− C

B , D

is

the

mid

poin

t of

−−

AC

.P

rove

: �A

BD

� �

CB

D

Pro

of:

Sta

tem

en

ts

Reaso

ns

1.

−−

A

B �

−−

C

B ,

D i

s t

he m

idp

oin

t o

f −

−

A

C 1

. G

iven

2.

−−

A

D �

−−

D

C

2.

Defi

nit

ion

of

Mid

po

int

3.

−−

B

D �

−−

B

D

3.

Refl

exiv

e P

rop

ert

y o

f C

on

gru

en

ce

4.

�A

BD

� �

CB

D

4.

SS

S

T

R

P

F

E

D

4-4

Skill

s Pr

acti

ceP

rovin

g T

rian

gle

s C

on

gru

en

t—S

SS

, S

AS

AB

= 2

, K

L =

2,

BC

= 2

√

2 ,

LM

= 2

√

2 ,

AC

= 2

, K

M =

2.

Th

e c

orr

esp

on

din

g s

ides h

ave t

he s

am

e m

easu

re a

nd

are

co

ng

ruen

t,

so

�A

BC

� �

KLM

by S

SS

.

AB

= 3

, K

L =

3,

BC

= √

13 ,

LM

= 4

, A

C =

√

10 ,

KM

= 5

.

Th

e c

orr

esp

on

din

g s

ides a

re n

ot

co

ng

ruen

t, s

o �

AB

C i

s n

ot

co

ng

ruen

t to

�K

LM

.

Pro

of:

019_

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M




An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

28

G

lenc

oe G

eom

etry

Prac

tice

Pro

vin

g T

rian

gle

s C

on

gru

en

t—S

SS

, S

AS

Det

erm

ine

wh

eth

er �

DE

F �

�P

QR

giv

en 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

)

2. D

(-7,

-3)

, E(-

4, -

1), F

(-2,

-5)

, P(2

, -2)

, Q(5

, -4)

, R(0

, -5)

3. W

rite

a f

low

pro

of.

Giv

en:

−−

RS

� −

−

TS

V

is

the

mid

poin

t of

−−

RT

.P

rove

: �

RS

V �

�T

SV

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

it

is n

ot p

ossi

ble

to

pro

ve c

ongr

uen

ce, w

rite

not

pos

sibl

e.

4.

5.

6.

7. I

ND

IREC

T M

EASU

REM

ENT

To

mea

sure

th

e w

idth

of

a si

nkh

ole

on

h

is p

rope

rty,

Har

mon

mar

ked

off

con

gru

ent

tria

ngl

es a

s sh

own

in

th

e di

agra

m. H

ow d

oes

he

know

th

at t

he

len

gth

s A

'B' a

nd

AB

are

equ

al?

A'

B'

AB

C

S

R V T

4-4 DE

= 5

√ �

2 ,

PQ

= 5

√ �

2 ,

EF

= 2

√ �

�

10 ,

QR

= 2

√ �

�

10 ,

DF =

5 √

�

2 ,

PR

= 5

√ �

2 .

�D

EF

� �

PQ

R b

y S

SS

sin

ce c

orr

esp

on

din

g s

ides h

ave t

he s

am

e

measu

re a

nd

are

co

ng

ruen

t.

DE

= √

��

13 ,

PQ

= √

��

13 ,

EF =

2 √

�

5 ,

QR

= √

��

26 ,

DF =

√ �

�

29 ,

PR

= √

��

13 .

Co

rresp

on

din

g s

ides a

re n

ot

co

ng

ruen

t, s

o �

DE

F i

s n

ot

co

ng

ruen

t to

�P

QR

.

Sin

ce ∠

AC

B a

nd

∠A

'CB

' are

vert

ical

an

gle

s,

they a

re c

on

gru

en

t. I

n t

he

fig

ure

, −

−

A

C �

−−

−

A

′C

an

d −

−

B

C �

−−

−

B

'C

. S

o �

AB

C �

�

A'B

'C

by S

AS

. B

y C

PC

TC

, th

e l

en

gth

s A

'B

' a

nd

AB

are

eq

ual.

Pro

of:

n

ot

po

ssib

le

SA

S o

r S

SS

S

SS

RS

�TS

Give

n

SV�

SVRe

flexi

vePr

oper

ty

RV

�V

TDe

finiti

onof

mid

poin

t

V is

th

em

idp

oin

t o

f R

T.Gi

ven

ΔR

SV�

ΔTS

VSS

S

019_

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M


NA

ME

DA

TE

PE

RIO

D

Lesson 4-4

Cha

pte

r 4

29

G

lenc

oe G

eom

etry

Wor

d Pr

oble

m P

ract

ice

Pro

vin

g T

rian

gle

s C

on

gru

en

t—S

SS

, S

AS

1. S

TIC

KS

Tys

on h

ad t

hre

e st

icks

of

len

gth

s 24

in

ches

, 28

inch

es, a

nd

30 i

nch

es. I

s it

pos

sibl

e to

mak

e tw

o n

onco

ngr

uen

t tr

ian

gles

usi

ng

the

sam

e th

ree

stic

ks?

Exp

lain

.

2. B

AK

ERY

Son

ia m

ade

a sh

eet

of

bakl

ava.

Sh

e h

as m

arki

ngs

on

her

pan

so

th

at s

he

can

cu

t th

em i

nto

lar

ge

squ

ares

. Aft

er s

he

cuts

th

e pa

stry

in

sq

uar

es, s

he

cuts

th

em d

iago

nal

ly t

o fo

rm t

wo

con

gru

ent

tria

ngl

es. W

hic

h

post

ula

te c

ould

you

use

to

prov

e th

e tw

o tr

ian

gles

con

gru

ent?

3. C

AK

E C

arl

had

a p

iece

of

cake

in

th

e sh

ape

of a

n i

sosc

eles

tri

angl

e w

ith

an

gles

26,

77,

an

d 77

. He

wan

ted

to

divi

de i

t in

to t

wo

equ

al p

arts

, so

he

cut

it t

hro

ugh

th

e m

iddl

e of

th

e 26

an

gle

to

the

mid

poin

t of

th

e op

posi

te s

ide.

He

says

th

at b

ecau

se h

e is

div

idin

g it

at

the

mid

poin

t of

a s

ide,

th

e tw

o pi

eces

are

co

ngr

uen

t. I

s th

is e

nou

gh i

nfo

rmat

ion

? E

xpla

in.

4. T

ILES

Tam

my

inst

alls

bat

hro

om t

iles

. H

er c

urr

ent

job

requ

ires

til

es t

hat

are

eq

uil

ater

al t

rian

gles

an

d al

l th

e ti

les

hav

e to

be

con

gru

ent

to e

ach

oth

er. S

he

has

a b

ig s

ack

of t

iles

all

in

th

e sh

ape

of e

quil

ater

al t

rian

gles

. Alt

hou

gh s

he

know

s th

at a

ll t

he

tile

s ar

e eq

uil

ater

al,

she

is n

ot s

ure

th

ey a

re a

ll t

he

sam

e si

ze. W

hat

mu

st s

he

mea

sure

on

eac

h

tile

to

be s

ure

th

ey a

re c

ongr

uen

t?

Exp

lain

.

5. I

NV

ESTI

GA

TIO

NA

n i

nve

stig

ator

at

a cr

ime

scen

e fo

un

d a

tria

ngu

lar

piec

e of

to

rn f

abri

c. T

he

inve

stig

ator

re

mem

bere

d th

at o

ne

of t

he

susp

ects

h

ad a

tri

angu

lar

hol

e in

th

eir

coat

. P

erh

aps

it w

as a

mat

ch. U

nfo

rtu

nat

ely,

to

avo

id t

ampe

rin

g w

ith

evi

den

ce, t

he

inve

stig

ator

did

not

wan

t to

tou

ch t

he

fabr

ic a

nd

cou

ld n

ot f

it i

t to

th

e co

at

dire

ctly

.

a. I

f th

e in

vest

igat

or m

easu

res

all

thre

e si

de l

engt

hs

of t

he

fabr

ic a

nd

the

hol

e, c

an t

he

inve

stig

ator

mak

e a

con

clu

sion

abo

ut

wh

eth

er o

r n

ot t

he

hol

e co

uld

hav

e be

en f

ille

d by

th

e fa

bric

?

b.

If t

he

inve

stig

ator

mea

sure

s tw

o si

des

of t

he

fabr

ic a

nd

the

incl

ude

d an

gle

and

then

mea

sure

s tw

o si

des

of t

he

hol

e an

d th

e in

clu

ded

angl

e ca

n h

e de

term

ine

if i

t is

a m

atch

? E

xpla

in.

4-4

No

. T

he s

ticks d

o n

ot

ch

an

ge

siz

e,

so

an

y a

rran

gem

en

t w

ill

yie

ld a

tri

an

gle

co

ng

ruen

t to

th

e

on

e b

efo

re.

eit

her

SS

S o

r S

AS

Sam

ple

an

sw

er:

No

. T

hat

is

on

ly e

xp

lain

ing

on

e s

ide t

hat

is

co

ng

ruen

t o

n t

he t

rian

gle

s.

He

need

s t

o s

ho

w t

wo

mo

re s

ides

of

the t

rian

gle

are

co

ng

ruen

t to

p

rove t

he p

ieces a

re c

on

gru

en

t.

yes

Sh

e j

ust

need

s t

o m

easu

re o

ne

sid

e o

f each

tile b

ecau

se t

hey

are

all e

qu

ilate

ral

tria

ng

les.

On

ly i

f th

e t

wo

sid

es a

nd

in

clu

ded

an

gle

of

on

e t

rian

gle

are

th

e c

orr

esp

on

din

g s

ides

an

d i

nclu

ded

an

gle

of

the

oth

er

tria

ng

le.

019_

036_

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47A

M




Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

30

G

lenc

oe G

eom

etry

Enri

chm

ent

4-4

Co

ng

ruen

t Tri

an

gle

s i

n t

he C

oo

rdin

ate

Pla

ne

If y

ou k

now

th

e co

ordi

nat

es o

f th

e ve

rtic

es o

f tw

o tr

ian

gles

in

th

e co

ordi

nat

e pl

ane,

you

can

oft

en d

ecid

e w

het

her

th

e tw

o tr

ian

gles

are

con

gru

ent.

Th

ere

may

be

mor

e th

an o

ne

way

to

do t

his

.

1. C

onsi

der

�A

BD

an

d �

CD

B w

hos

e ve

rtic

es h

ave

coor

din

ates

A(0

, 0),

B

(2, 5

), C

(9, 5

), a

nd

D(7

, 0).

Bri

efly

des

crib

e h

ow y

ou c

an u

se w

hat

you

kn

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

.

2. C

onsi

der

�P

QR

an

d �

KL

M w

hos

e ve

rtic

es a

re t

he

foll

owin

g po

ints

.

P

(1, 2

) Q

(3, 6

) R

(6, 5

)

K

(-2,

1)

L(-

6, 3

) M

(-5,

6)

B

rief

ly d

escr

ibe

how

you

can

sh

ow t

hat

�P

QR

� �

KL

M.

3. I

f yo

u k

now

th

e co

ordi

nat

es o

f al

l th

e ve

rtic

es o

f tw

o tr

ian

gles

, is

it

alw

ays

poss

ible

to

tell

wh

eth

er t

he

tria

ngl

es a

re c

ongr

uen

t? E

xpla

in.

Sam

ple

an

sw

er:

Sh

ow

th

at

the s

lop

es o

f −

−

A

D a

nd

−−

B

C a

re e

qu

al. C

on

clu

de

that

−−

B

C |

| −

−

A

D .

Use t

he a

ng

le r

ela

tio

nsh

ips f

or

para

llel

lin

es a

nd

a

tran

svers

al

to c

on

clu

de t

hat

∠C

BD

is c

on

gru

en

t to

∠B

DA

. C

ou

nt

the

len

gth

s o

f th

e h

ori

zo

nta

l seg

men

ts t

o c

on

clu

de t

hat

BC

= A

D.

Usin

g

these f

acts

an

d t

he f

act

that

−−

B

D i

s a

co

mm

on

sid

e f

or

the t

rian

gle

s t

o

co

nclu

de t

hat

�A

BD

� �

CD

B b

y S

AS

.

Use t

he D

ista

nce F

orm

ula

to

fin

d t

he l

en

gth

s o

f th

e s

ides o

f b

oth

tr

ian

gle

s.

Co

nclu

de t

hat

�P

QR

� �

KLM

by S

SS

.

Yes;

yo

u c

an

use t

he D

ista

nce F

orm

ula

an

d S

SS

.

019_

036_

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OC

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C04

_890

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812

:32:

51A

M


NA

ME

DA

TE

PE

RIO

D

Lesson 4-5

Cha

pte

r 4

31

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

Pro

vin

g T

rian

gle

s C

on

gru

en

t—A

SA

, A

AS

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

ar

e co

ngr

uen

t.

W

rite

a t

wo

colu

mn

pro

of.

Giv

en:

−−

AB

‖ −−

− C

D

∠

CB

D �

∠A

DB

Pro

ve: �

AB

D �

�C

DB

Sta

tem

ents

Rea

son

s1.

−−

AB

‖ −−

− C

D

2. ∠

CB

D �

∠A

DB

3. ∠

AB

D �

∠B

DC

4. B

D =

BD

5. �

AB

D �

�C

DB

1. G

iven

2. G

iven

3. A

lter

nat

e In

teri

or A

ngl

es T

heo

rem

4. R

efle

xive

Pro

pert

y of

con

gru

ence

5. A

SA

4-5

AS

A P

os

tula

teIf

tw

o a

ng

les a

nd

th

e in

clu

de

d s

ide

of

on

e t

ria

ng

le a

re c

on

gru

en

t to

tw

o a

ng

les a

nd

the

in

clu

de

d s

ide

of

an

oth

er

tria

ng

le,

the

n t

he

tria

ng

les a

re c

on

gru

en

t.

Exam

ple

Exer

cise

sPR

OO

F W

rite

th

e sp

ecif

ied

typ

e of

pro

of.

1. W

rite

a t

wo

colu

mn

pro

of.

2. W

rite

a p

arag

raph

pro

of.

S

V

U

RT

AC

B

E

D

Giv

en: ∠

S �

∠V

,

G

iven

: −−

− C

D b

isec

ts −

−

AE

, −

−

AB

‖ −−

− C

D

T

is

the

mid

poin

t of

−−

SV

∠

E �

∠B

CA

Pro

ve: �

RT

S �

�U

TV

P

rove

: �A

BC

� �

CD

E

Pro

of:

W

e k

no

w t

hat

∠E

� ∠

BC

A

an

d −−

− C

D b

isects

−−

AE

. S

ince −−

− C

D

bis

ects

−−

AE

, b

y t

he d

efi

nit

ion

of

bis

ecto

r, A

C =

CE

. W

e a

re a

lso

g

iven

th

at

−−−

AB

‖ −−

− C

D ,

fro

m t

his

we

can

dete

rmin

e t

hat

∠A

is

co

ng

ruen

t to

∠D

CE

by t

he

Alt

ern

ate

In

teri

or

An

gle

T

heo

rem

. F

rom

th

is w

e k

no

w

that

∠A

BC

�

∠C

DE

by t

he

AS

A T

heo

rem

.

Sta

tem

en

tsR

easo

ns

1.

∠S

� ∠

V

T i

s t

he

mid

po

int

of

−−

S

V

2.

ST

= T

V

3.

∠R

TS

� ∠

VTU

4.

�R

TS

� �

UTV

1.

Giv

en

2.

defi

nit

ion

of

Mid

po

int

3.

Vert

ical

An

gle

th

eo

rem

.

4.

AS

A

019_

036_

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:32:

54A

M




An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

32

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Pro

vin

g T

rian

gle

s C

on

gru

en

t—A

SA

, A

AS

AA

S Th

eore

m A

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)

Th

eore

m.

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

• A

SA

Pos

tula

te•

SS

S P

ostu

late

•

AA

S T

heo

rem

• S

AS

Pos

tula

te

In

th

e d

iagr

am, ∠

BC

A �

∠D

CA

. Wh

ich

sid

es

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 T

heo

rem

?

−−

AC

� −

−

AC

by

the

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, b

ecau

se −

−

AC

wou

ld b

e th

e in

clu

ded

side

. If

∠B

� ∠

D, t

hen

�A

BC

� �

AD

C b

y th

e A

AS

Th

eore

m.

Exer

cise

sPR

OO

F W

rite

th

e sp

ecif

ied

typ

e of

pro

of.

1. W

rite

a t

wo

colu

mn

pro

of.

G

iven

: −−

− B

C ‖

−−

EF

AB

= E

D

∠C

� ∠

F

Pro

ve: �

AB

D �

�D

EF

Sta

tem

en

tsR

easo

ns

1.

−−

B

C ‖

−−

E

F

A

B =

ED

∠

C �

∠F

2.

∠A

BD

� ∠

DE

F

3.

�A

BD

� �

DE

F

1.

Giv

en

2.

Co

rresp

on

din

g A

ng

les T

heo

rem

3.

AA

S

2. W

rite

a f

low

pro

of.

G

iven

: ∠S

� ∠

U;

−−

TR

bis

ects

∠S

TU

.

Pro

ve: ∠

SR

T �

∠U

RT

P

roo

f:

S

RT

U

4-5

AA

S T

he

ore

mIf

tw

o a

ng

les a

nd

a n

on

inclu

de

d s

ide

of

on

e t

ria

ng

le a

re c

on

gru

en

t to

th

e c

orr

esp

on

din

g

two

an

gle

s a

nd

sid

e o

f a

se

co

nd

tria

ng

le,

the

n t

he

tw

o t

ria

ng

les a

re c

on

gru

en

t.

Exam

ple

D

C1 2

A

B

Give

n

Give

n

RT

�R

T

Refl.

Pro

p. o

f �

Def.o

f∠ b

isec

tor

TR b

isec

ts ∠

STU

.

SRT

� U

RT

∠ST

R�

∠U

TR

AAS

∠SR

T�

∠U

RT

CPCT

C∠

S�

∠U

019_

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:33:

01A

M


NA

ME

DA

TE

PE

RIO

D

Lesson 4-5

Cha

pte

r 4

33

G

lenc

oe G

eom

etry

PRO

OF

Wri

te a

flo

w p

roof

.

1. G

iven

: ∠

N �

∠L

−

−

JK

� −

−−

MK

Pro

ve:

�J

KN

� �

MK

L

2. G

iven

: −

−

AB

� −−

− C

B

∠

A �

∠C

−−

− D

B b

isec

ts ∠

AB

C.

P

rove

: −−

− A

D �

−−−

CD

3. W

rite

a p

arag

raph

pro

of.

G

iven

: −−

− D

E ‖

−−−

FG

∠

E �

∠G

P

rove

: �

DF

G �

�F

DE

FG

DE

AC

B D

N

J

M

KL

4-5

Skill

s Pr

acti

ceP

rovin

g T

rian

gle

s C

on

gru

en

t—A

SA

, A

AS

Pro

of:

P

roo

f: S

ince i

t is

giv

en

th

at

−−

D

E ‖

−

−

FG

, it

fo

llo

ws t

hat

∠E

DF �

∠G

FD

,

b

ecau

se a

lt.

int.

� o

f p

ara

llel

lin

es a

re �

. It

is g

iven

th

at

∠E

� ∠

G.

By t

he

Refl

exiv

e

P

rop

ert

y,

−−

D

F �

−−

FD

. S

o �

DFG

� �

FD

E b

y A

AS

.

∠N

�∠

LGi

ven

JK�

MK

Give

n

∠JK

N�

∠M

KL

Verti

cal �

are

�.

�JK

N�

�M

KL

AAS

Pro

of:

∠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

019_

036_

GE

OC

RM

C04

_890

513.

indd

335/

22/0

88:

05:4

0P

M




Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

34

G

lenc

oe G

eom

etry

PRO

OF

Wri

te t

he

spec

ifie

d t

ype

of p

roof

. 1

. Wri

te a

flo

w p

roof

.

Giv

en:

S i

s th

e m

idpo

int

of −−

−Q

T .

−−−

QR

‖ −−

− T

U

P

rove

: �

QS

R �

�T

SU

2. W

rite

a p

arag

raph

pro

of.

G

iven

: ∠

D �

∠F

−−−

GE

bis

ects

∠D

EF

.

Pro

ve:

−−−

DG

� −−

− F

G

AR

CH

ITEC

TUR

E F

or E

xerc

ises

3 a

nd

4, u

se t

he

foll

owin

g

info

rmat

ion

.A

n a

rch

itec

t u

sed

the

win

dow

des

ign

in

th

e di

agra

m w

hen

rem

odel

ing

an a

rt s

tudi

o. −

−

AB

an

d −−

− C

B e

ach

mea

sure

3 f

eet.

3. S

upp

ose

D i

s th

e m

idpo

int

of −

−

AC

. Det

erm

ine

wh

eth

er �

AB

D �

�C

BD

. Ju

stif

y yo

ur

answ

er.

4. S

upp

ose

∠A

� ∠

C. D

eter

min

e w

het

her

�A

BD

� �

CB

D. J

ust

ify

you

r an

swer

.DB

AC

D

G

F

E

UQ

S

RT

4-5

Prac

tice

Pro

vin

g T

rian

gle

s C

on

gru

en

t—A

SA

, A

AS

Pro

of:

Pro

of:

Sin

ce i

t is

giv

en

th

at

−−

G

E b

isects

∠D

EF,

∠D

EG

� ∠

FE

G b

y t

he

defi

nit

ion

of

an

an

gle

bis

ecto

r. I

t is

giv

en

th

at

∠D

� ∠

F.

By t

he

Refl

exiv

e P

rop

ert

y,

−−

G

E �

−−

G

E .

So

�D

EG

� �

FE

G b

y A

AS

. T

here

fore

−−

D

G �

−−

FG

by C

PC

TC

.

Sin

ce D

is t

he m

idp

oin

t o

f −

−

A

C ,

−−

A

D �

−−

C

D b

y t

he M

idp

oin

t T

heo

rem

.

−−

A

B �

−−

C

B b

y t

he d

efi

nit

ion

of

co

ng

ruen

t seg

men

ts.

By t

he R

efl

exiv

e

Pro

pert

y −

−

B

D �

−−

B

D .

So

�A

BD

� �

CB

D b

y S

SS

.

We a

re g

iven

−−

A

B �

−−

C

B a

nd

∠A

� ∠

C.

−−

B

D �

−−

B

D b

y t

he R

efl

exiv

e

Pro

pert

y.

Sin

ce S

SA

can

no

t b

e u

sed

to

pro

ve t

hat

tria

ng

les a

re

co

ng

ruen

t, w

e c

an

no

t say w

heth

er

�A

BD

� �

CB

D.

∠Q

�∠

T

Give

n

QR

||

TUGi

ven

Mid

poin

t The

orem

Alt.

Int.

� a

re �

.

QS

�TS

S is

th

e m

idp

oin

t o

f Q

T.

QSR

�

TSU

ASA

∠Q

SR�

∠TS

UVe

rtica

l � a

re �

.

019_

036_

GE

OC

RM

C04

_890

513.

indd

344/

12/0

812

:33:

12A

M


NA

ME

DA

TE

PE

RIO

D

Lesson 4-5

Cha

pte

r 4

35

G

lenc

oe G

eom

etry

1. D

OO

R S

TOPS

Tw

o do

or s

tops

hav

e cr

oss-

sect

ion

s th

at a

re r

igh

t tr

ian

gles

. T

hey

bot

h h

ave

a 20

an

gle

and

the

len

gth

of

the

side

bet

wee

n t

he

90 a

nd

20 a

ngl

es a

re e

qual

. Are

th

e cr

oss-

sect

ion

s co

ngr

uen

t? E

xpla

in.

2. M

APP

ING

Tw

o pe

ople

dec

ide

to t

ake

a w

alk.

On

e pe

rson

is

in B

omba

y an

d th

e ot

her

is

in M

ilw

auke

e. T

hey

sta

rt b

y w

alki

ng

stra

igh

t fo

r 1

kilo

met

er. T

hen

th

ey b

oth

tu

rn r

igh

t at

an

an

gle

of 1

10,

and

con

tin

ue

to w

alk

stra

igh

t ag

ain

. A

fter

a w

hil

e, t

hey

bot

h t

urn

rig

ht

agai

n, b

ut

this

tim

e at

an

an

gle

of 1

20.

Th

ey e

ach

wal

k st

raig

ht

for

a w

hil

e in

th

is n

ew d

irec

tion

un

til

they

en

d u

p w

her

e th

ey s

tart

ed. E

ach

per

son

wal

ked

in a

tri

angu

lar

path

at

thei

r lo

cati

on.

Are

th

ese

two

tria

ngl

es c

ongr

uen

t?

Exp

lain

.

3. C

ON

STR

UC

TIO

N T

he

roof

top

of

An

gelo

’s h

ouse

cre

ates

an

equ

ilat

eral

tr

ian

gle

wit

h t

he

atti

c fl

oor.

An

gelo

w

ants

to

divi

de h

is a

ttic

in

to 2

equ

al

part

s. H

e th

inks

he

shou

ld d

ivid

e it

by

plac

ing

a w

all

from

th

e ce

nte

r of

th

e ro

of

to t

he

floo

r at

a 9

0 an

gle.

If

An

gelo

doe

s th

is, t

hen

eac

h s

ecti

on w

ill

shar

e a

side

an

d h

ave

corr

espo

ndi

ng

90 a

ngl

es. W

hat

el

se m

ust

be

expl

ain

ed t

o pr

ove

that

th

e tw

o tr

ian

gula

r se

ctio

ns

are

con

gru

ent?

90˚

90˚

4. L

OG

IC W

hen

Car

olyn

fin

ish

ed h

er

mu

sica

l tr

ian

gle

clas

s, h

er t

each

er g

ave

each

stu

den

t in

th

e cl

ass

a ce

rtif

icat

e in

th

e sh

ape

of a

gol

den

tri

angl

e. E

ach

st

ude

nt

rece

ived

a d

iffe

ren

t sh

aped

tr

ian

gle.

Car

olyn

los

t h

er t

rian

gle

on h

er

way

hom

e. L

ater

sh

e sa

w p

art

of a

go

lden

tri

angl

e u

nde

r a

grat

e.

Is e

nou

gh o

f th

e tr

ian

gle

visi

ble

to a

llow

C

arol

yn t

o de

term

ine

that

th

e tr

ian

gle

is

inde

ed h

ers?

Exp

lain

.

5. P

AR

K M

AIN

TEN

AN

CE

Par

k of

fici

als

nee

d a

tria

ngu

lar

tarp

to

cove

r a

fiel

d sh

aped

lik

e an

equ

ilat

eral

tri

angl

e 20

0 fe

et o

n a

sid

e.

a. S

upp

ose

you

kn

ow t

hat

a t

rian

gula

r ta

rp h

as t

wo

60 a

ngl

es a

nd

one

side

of

len

gth

200

fee

t. W

ill

this

tar

p co

ver

the

fiel

d? E

xpla

in.

b.

Su

ppos

e yo

u k

now

th

at a

tri

angu

lar

tarp

has

th

ree

60 a

ngl

es. W

ill

this

ta

rp n

eces

sari

ly c

over

th

e fi

eld?

E

xpla

in.

4-5

Wor

ld P

robl

em P

rati

ceP

rovin

g T

rian

gle

s C

on

gru

en

t—A

SA

, A

AS

yes,

by A

SA

Sam

ple

an

sw

er:

Becau

se t

he

ori

gin

al

tria

ng

le w

as e

qu

ilate

ral,

on

e m

ore

pair

of

an

gle

s i

s

co

ng

ruen

t. T

he t

rian

gle

s a

re

co

ng

ruen

t b

y A

AS

.

No

; th

ere

are

eq

uia

ng

ula

r

tria

ng

les o

f all s

izes.

yes,

by A

AS

Yes,

if s

he k

no

ws t

he m

easu

res

of

her

lost

tria

ng

le.

Sin

ce t

wo

an

gle

s a

nd

th

e i

nclu

ded

sid

e

betw

een

th

em

can

be s

een

, sh

e

can

dete

rmin

e i

f it

is c

on

gru

en

t b

y t

he A

SA

Th

eo

rem

.

Yes,

becau

se t

wo

kn

ow

n

an

gle

s a

re 6

0,

the t

hir

d m

ust

be a

lso

. A

SA

or

AA

S c

an

be

used

to

sh

ow

th

at

it w

ill

co

ver.

019_

036_

GE

OC

RM

C04

_890

513.

indd

354/

12/0

812

:33:

17A

M




An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass


NA

ME

DA

TE

PE

RIO

D

Cha

pte

r 4

36

G

lenc

oe G

eom

etry

Enri

chm

ent

4-5

Wh

at

Ab

ou

t A

AA

to

Pro

ve T

rian

gle

Co

ng

ruen

ce?

You

hav

e le

arn

ed a

bou

t th

e S

SS

an

d A

SA

pos

tula

tes

to p

rove

th

at t

wo

tria

ngl

es a

re

con

gru

ent.

Do

you

th

ink

that

tri

angl

es c

an b

e pr

oven

con

gru

ent

by A

AA

?

Inve

stig

ate

the

AA

S t

heo

rem

. AA

S r

epre

sen

ts t

he

con

ject

ure

th

at t

wo

tria

ngl

es a

re c

ongr

uen

t if

th

ey h

ave

two

pair

s of

co

rres

pon

din

g an

gles

con

gru

ent

and

corr

espo

ndi

ng

non

-in

clu

ded

side

s co

ngr

uen

t. F

or e

xam

ple,

in

th

e tr

ian

gles

at

the

righ

t, ∠

A �

∠Y

, ∠B

� ∠

X, a

nd

−−

AC

� −

−

YZ

.

1. R

efer

to

the

draw

ing

abov

e. W

hat

is

the

mea

sure

of

the

mis

sin

g an

gle

in t

he

firs

t tr

ian

gle?

Wh

at i

s th

e m

easu

re o

f th

e m

issi

ng

angl

e in

th

e se

con

d tr

ian

gle?

2. W

ith

th

e in

form

atio

n f

rom

Exe

rcis

e 1,

wh

at p

ostu

late

can

you

now

use

to

prov

e th

at t

he

two

tria

ngl

es a

re c

ongr

uen

t?

Now

in

vest

igat

e th

e A

AA

con

ject

ure

. AA

A r

epre

sen

ts t

he

con

ject

ure

th

at t

wo

tria

ngl

es a

re

con

gru

ent

if t

hey

hav

e th

ree

pair

s of

cor

resp

ondi

ng

angl

es c

ongr

uen

t. T

he

tria

ngl

es b

elow

de

mon

stra

te A

AA

.

25˚

25˚

70˚

70˚

85˚

85˚

3. A

re t

he

tria

ngl

es a

bove

con

gru

ent?

Exp

lain

.

4. C

an y

ou d

raw

tw

o tr

ian

gles

th

at h

ave

all

thre

e an

gles

con

gru

ent

that

are

not

con

gru

ent?

C

an A

AA

be

use

d to

pro

ve t

hat

tw

o tr

ian

gles

are

con

gru

ent?

5. W

hat

ch

arac

teri

stic

s do

th

ese

two

tria

ngl

es d

raw

n b

y A

AA

see

m t

o po

sses

s?

89˚

89˚

66˚

66˚

A

BC

5 in

5 in

X

Y

Z

25;

25

AS

A

No

, th

e s

ides h

ave d

iffe

ren

t le

ng

ths.

yes;

no

Th

ey h

ave t

he s

am

e s

hap

e,

bu

t d

iffe

ren

t siz

es.

019_

036_

GE

OC

RM

C04

_890

513.

indd

364/

12/0

812

:33:

25A

M

NA

ME

DA

TE

PE

RIO

D


Lesson 4-6

Cha

pte

r 4

37

G

lenc

oe G

eom

etry

Exer

cise

sA

LGEB

RA

Fin

d t

he

valu

e of

eac

h v

aria

ble

.

1.

R

P

Q2x

°

40°

2.

S

V

T3x

- 6

2x+

6

3.

W

YZ

3x°

4.

2x°

(6x

+6)

°

5.

5x°

6.

RS

T 3x°

x°

7. P

RO

OF

Wri

te a

tw

o-co

lum

n p

roof

.

G

iven

: ∠1

� ∠

2

Pro

ve:

−−

AB

� −−

− C

B

Pro

of:

Sta

tem

en

ts

Reaso

ns

Stud

y G

uide

and

Inte

rven

tion

Iso

scele

s a

nd

Eq

uilate

ral

Tri

an

gle

s

Pro

per

ties

of

Iso

scel

es T

rian

gle

s A

n i

sosc

eles

tri

angl

e h

as t

wo

con

gru

ent

side

s ca

lled

th

e le

gs. T

he

angl

e fo

rmed

by

the

legs

is

call

ed t

he

vert

ex a

ngl

e. T

he

oth

er t

wo

angl

es a

re c

alle

d b

ase

angl

es. Y

ou c

an p

rove

a t

heo

rem

an

d it

s co

nve

rse

abou

t is

osce

les

tria

ngl

es.

•

If t

wo

side

s of

a t

rian

gle

are

con

gru

ent,

th

en t

he

angl

es o

ppos

ite

th

ose

side

s ar

e co

ngr

uen

t. (

Isos

cele

s T

rian

gle

Th

eore

m)

•

If t

wo

angl

es o

f a

tria

ngl

e ar

e co

ngr

uen

t, t

hen

th

e si

des

oppo

site

th

ose

angl

es a

re c

ongr

uen

t. (

Con

vers

e of

Iso

scel

es T

rian

gle

Th

eore

m)

If −

−

AB

� −−

− C

B ,

then ∠

A �

∠C

.

If ∠

A �

∠C

, th

en −

−

AB

� −−

− C

B .

A

B

C

F

ind

x, g

iven

−−

B

C �

−−

B

A .

B

AC( 4

x+

5) °

( 5x

- 1

0)°

BC

= B

A, s

o

m∠

A =

m∠

C

Isos.

Triangle

Theore

m

5x

- 1

0 =

4x

+ 5

S

ubstitu

tion

x

- 1

0 =

5

subtr

act

4x

from

each s

ide.

x

= 1

5 A

dd 1

0 t

o e

ach s

ide.

Exam

ple

1Ex

amp

le 2

F

ind

x.

RT

S 3x-

13

2x

m∠

S =

m∠

T, s

o

SR

= T

R

Convers

e o

f Is

os. �

Thm

.

3x

- 1

3 =

2x

Substitu

tion

3x

= 2

x +

13

Add 1

3 t

o e

ach s

ide.

x

= 1

3 S

ubtr

act

2x

from

each s

ide.

B

AC

D

E

13

2

4-6

1.

∠1 �

∠2

1. G

iven

2.

∠2 �

∠3

2. V

ert

ical

an

gle

s a

re c

on

gru

en

t.3.

∠1 �

∠3

3. T

ran

sit

ive P

rop

ert

y o

f �

4.

−−

A

B �

−−

C

B

4. C

on

vers

e o

f Is

os.

Tri

an

gle

Th

m.

12

936

35

12

15

037_

054_

GE

OC

RM

C04

_890

513.

indd

374/

12/0

812

:35:

32A

M




Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass

NA

ME

DA

TE

PE

RIO

D


Cha

pte

r 4

38

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Iso

scele

s a

nd

Eq

uilate

ral

Tri

an

gle

s

Pro

per

ties

of

Equ

ilate

ral T

rian

gle

s A

n e

qu

ilat

eral

tri

angl

e h

as t

hre

e co

ngr

uen

t si

des.

Th

e Is

osce

les

Tri

angl

e T

heo

rem

lea

ds t

o tw

o co

roll

arie

s ab

out

equ

ilat

eral

tri

angl

es.

1.

A t

ria

ng

le is e

qu

ilate

ral if a

nd

on

ly if

it is e

qu

ian

gu

lar.

2.

Ea

ch

an

gle

of

an

eq

uila

tera

l tr

ian

gle

me

asu

res 6

0°.

P

rove

th

at i

f a

lin

e is

par

alle

l to

on

e si

de

of a

n

equ

ilat

eral

tri

angl

e, t

hen

it

form

s an

oth

er e

qu

ilat

eral

tri

angl

e.

Pro

of:

Sta

tem

ents

R

easo

ns

1. �

AB

C i

s eq

uil

ater

al; −−

− P

Q ‖

−−−

BC

. 1.

Giv

en2.

m∠

A =

m∠

B =

m∠

C =

60

2. E

ach

∠ o

f an

equ

ilat

eral

� m

easu

res

60°.

3. ∠

1 �

∠B

, ∠2

� ∠

C

3. I

f ‖

lin

es, t

hen

cor

res.

� a

re �

.4.

m∠

1 =

60,

m∠

2 =

60

4. S

ubs

titu

tion

5. �

AP

Q i

s eq

uil

ater

al.

5.

If a

� i

s eq

uia

ngu

lar,

th

en i

t is

equ

ilat

eral

.

Exer

cise

s

ALG

EBR

A F

ind

th

e va

lue

of e

ach

var

iab

le.

1.

D

FE

6x°

2.

G

JH

6x-

55x

3.

3x°

4.

4x40

60°

5.

X

ZY

4x-

4

3x+

860

°

6.

4x°

60°

7. P

RO

OF

Wri

te a

tw

o-co

lum

n p

roof

.

G

iven

: �A

BC

is

equ

ilat

eral

; ∠1

� ∠

2.

P

rove

: ∠A

DB

� ∠

CD

B

P

roo

f:

S

tate

men

ts

Reaso

ns

A D C

B1 2

A

B

PQ

C

12

4-6

Exam

ple

1

. �

AB

C i

s e

qu

ilate

ral.

1.

Giv

en

2

. −

−

A

B �

−−

C

B ;

∠A

� ∠

C

2.

An

eq

uilate

ral

� h

as �

sid

es a

nd

� a

ng

les.

3

. ∠

1 �

∠

2

3.

Giv

en

4

. �

AB

D �

�

CB

D

4.

AS

A P

ostu

late

5

. ∠

AD

B �

∠

CD

B

5.

CP

CT

C

10

12

15

10

520

037_

054_

GE

OC

RM

C04

_890

513.

indd

384/

12/0

812

:35:

43A

M

NA

ME

DA

TE

PE

RIO

D


Lesson 4-6

Cha

pte

r 4

39

G

lenc

oe G

eom

etry

Skill

s Pr

acti

ceIs

oscele

s a

nd

Eq

uilate

ral

Tri

an

gle

s

D

C

B

AE

Ref

er t

o th

e fi

gure

at

the

righ

t.

1. I

f −

−

AC

� −−

− A

D , n

ame

two

con

gru

ent

angl

es.

2. I

f −−

− B

E �

−−−

BC

, nam

e tw

o co

ngr

uen

t an

gles

.

3. I

f ∠

EB

A �

∠E

AB

, nam

e tw

o co

ngr

uen

t se

gmen

ts.

4. I

f ∠

CE

D �

∠C

DE

, nam

e tw

o co

ngr

uen

t se

gmen

ts.

Fin

d e

ach

mea

sure

. 5

. m∠

AB

C

6. m

∠E

DF

60°

40°

ALG

EBR

A F

ind

th

e va

lue

of e

ach

var

iab

le.

7.

2x+

43x

-10

8.

(2x

+3)

°

9. P

RO

OF

Wri

te a

tw

o-co

lum

n p

roof

.

G

iven

: −−

− C

D �

−−−

CG

−−−

DE

� −−

− G

F

P

rove

: −−

− C

E �

−−

CF

D E F G

C

4-6

∠A

CD

� ∠

CD

A

∠B

EC

� ∠

BC

E

−−

E

B �

−−

E

A

−−

C

E �

−

−

C

D

P

roo

f:

Sta

tem

en

ts

Reaso

ns

1

. −

−

C

D �

−−

C

G

1.

Giv

en

3

. −

−

D

E �

−−

G

F

3.

Giv

en

4

. �

CD

E �

�

CG

F

4.

SA

S

5

. −

−

C

E �

−

−

C

F

5.

CP

CT

C

2

. ∠

D �

∠

G

2.

If 2

sid

es o

f a �

are

�,

then

th

e �

op

po

sit

e

tho

se s

ides a

re �

.

60

70

14

21

037_

054_

GE

OC

RM

C04

_890

513.

indd

395/

22/0

88:

06:4

8P

M




An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass

NA

ME

DA

TE

PE

RIO

D


Cha

pte

r 4

40

G

lenc

oe G

eom

etry

Prac

tice

Iso

scele

s a

nd

Eq

uilate

ral

Tri

an

gle

s

Linc

oln

Haw

ks

E DBC

A12

3 4

UR

TV

S

Ref

er t

o th

e fi

gure

at

the

righ

t.

1. I

f −−

− R

V �

−−

RT

, nam

e tw

o co

ngr

uen

t an

gles

. ∠

RTV

� ∠

RV

T

2. I

f −

−

RS

� −

−

SV

, nam

e tw

o co

ngr

uen

t an

gles

. ∠

SV

R �

∠S

RV

3. I

f ∠

SR

T �

∠S

TR

, nam

e tw

o co

ngr

uen

t se

gmen

ts. −

−

S

T �

−−

S

R

4. I

f ∠

ST

V �

∠S

VT

, nam

e tw

o co

ngr

uen

t se

gmen

ts.

−−

S

T �

−−

S

V

Fin

d e

ach

mea

sure

.

5. m

∠K

ML

6.

m∠

HM

G

7. m

∠G

HM

8. I

f m

∠H

JM

= 1

45, f

ind

m∠

MH

J.

9. I

f m

∠G

= 6

7, f

ind

m∠

GH

M.

10. P

RO

OF

Wri

te a

tw

o-co

lum

n p

roof

.

G

iven

: −−

− D

E ‖

−−−

BC

∠

1 �

∠2

P

rove

: −

−

AB

� −

−

AC

11. S

POR

TS A

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. I

f 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.

4-6

50°

81,

81

P

roo

f:

S

tate

men

ts

R

easo

ns

1. G

iven

2. C

orr

. �

are

�

.

3. G

iven

4. C

on

gru

en

ce o

f �

is t

ran

sit

ive.

5. I

f 2 �

of

a �

are

�

, th

en

th

e s

id

e

op

po

sit

e t

ho

se s

ides a

re �

1. −

−

D

E ‖

−

−

B

C

2. ∠

1 �

∠

4

∠

2 �

∠

3

3. ∠

1 �

∠

2

4. ∠

3 �

∠

4

5. −

−

A

B �

−−

A

C

60

70

40

17.5

46

037_

054_

GE

OC

RM

C04

_890

513.

indd

404/

12/0

812

:36:

00A

M

NA

ME

DA

TE

PE

RIO

D


Lesson 4-6

Cha

pte

r 4

41

G

lenc

oe G

eom

etry

Wor

d Pr

oble

m P

ract

ice

Iso

scele

s a

nd

Eq

uilate

ral

Tri

an

gle

s

1

1. T

RIA

NG

LES

At

an a

rt s

upp

ly s

tore

, tw

o di

ffer

ent

tria

ngu

lar

rule

rs a

re a

vail

able

. O

ne

has

an

gles

45°

, 45°

, an

d 90

°. T

he

oth

er h

as a

ngl

es 3

0°, 6

0°, a

nd

90°.

W

hic

h t

rian

gle

is i

sosc

eles

?

2. R

ULE

RS

A f

olda

ble

rule

r h

as t

wo

hin

ges

that

div

ide

the

rule

r in

to t

hir

ds. I

f th

e en

ds a

re f

olde

d u

p u

nti

l th

ey t

ouch

, w

hat

kin

d of

tri

angl

e re

sult

s?

3. H

EXA

GO

NS

Juan

ita

plac

ed o

ne

end

of

each

of

6 bl

ack

stic

ks a

t a

com

mon

poi

nt

and

then

spa

ced

the

oth

er e

nds

eve

nly

ar

oun

d th

at p

oin

t. S

he

con

nec

ted

the

free

en

ds o

f th

e st

icks

wit

h l

ines

. Th

e re

sult

was

a r

egu

lar

hex

agon

. Th

is

con

stru

ctio

n s

how

s th

at a

reg

ula

r h

exag

on c

an b

e m

ade

from

six

con

gru

ent

tria

ngl

es. C

lass

ify

thes

e tr

ian

gles

. E

xpla

in.

4. P

ATH

S A

mar

ble

path

is

con

stru

cted

ou

t of

sev

eral

con

gru

ent

isos

cele

s tr

ian

gles

. Th

e ve

rtex

an

gles

are

all

20.

W

hat

is

the

mea

sure

of

angl

e 1

in t

he

figu

re?

5. B

RID

GES

Eve

ry d

ay, c

ars

driv

e th

rou

gh

isos

cele

s tr

ian

gles

wh

en t

hey

go

over

th

e L

eon

ard

Zak

im B

ridg

e in

Bos

ton

. Th

e te

n-l

ane

road

way

for

ms

the

base

s of

th

e tr

ian

gles

.

a. T

he

angl

e la

bele

d A

in

th

e pi

ctu

re

has

a m

easu

re o

f 67

. Wh

at i

s th

e m

easu

re o

f ∠

B?

b.

Wh

at i

s th

e m

easu

re o

f ∠

C?

c. N

ame

the

two

con

gru

ent

side

s.

the 4

5°-4

5°- 9

0° t

rian

gle

A

B

C

−−

A

B �

−

−

B

C

4-6 eq

uilate

ral

tria

ng

le

Sam

ple

an

sw

er:

Sin

ce t

wo

sid

es

are

co

ng

ruen

t, t

hey a

re i

so

scele

s.

Becau

se t

he s

ticks a

re s

paced

even

ly,

the v

ert

ex a

ng

les (

at

the

cen

ter)

are

all 3

60

−

6

or

60.

Th

us t

he

base a

ng

les m

ust

be

(18

0 -

60

) −

6

or

60.

Sin

ce a

ll a

ng

les o

f an

y o

ne

tria

ng

le i

n t

he f

igu

re a

re 6

0,

all

sid

es a

re e

qu

al

an

d t

he t

rian

gle

s

are

eq

uilate

ral.

80 4

6

67

037_

054_

GE

OC

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C04

_890

513.

indd

414/

12/0

812

:36:

06A

M




Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass

NA

ME

DA

TE

PE

RIO

D


Cha

pte

r 4

42

G

lenc

oe G

eom

etry

Enri

chm

ent

Tri

an

gle

Ch

allen

ges

Som

e pr

oble

ms

incl

ude

dia

gram

s. I

f yo

u a

re n

ot s

ure

how

to

solv

e th

e pr

oble

m, b

egin

by

usi

ng

the

give

n i

nfo

rmat

ion

. Fin

d th

e m

easu

res

of a

s m

any

angl

es a

s yo

u c

an, w

riti

ng

each

m

easu

re o

n t

he

diag

ram

. Th

is m

ay g

ive

you

m

ore

clu

es t

o th

e so

luti

on.

1. G

iven

: B

E =

BF

, ∠B

FG

� ∠

BE

F �

∠

BE

D, m

∠B

FE

= 8

2 an

d A

BF

G a

nd

BC

DE

eac

h h

ave

oppo

site

sid

es p

aral

lel

and

con

gru

ent.

F

ind

m∠

AB

C. A

GD

FE

CB

3. G

iven

: m

∠U

ZY

= 9

0, m

∠Z

WX

= 4

5,

�Y

ZU

� �

VW

X, U

VX

Y i

s a

squ

are

(all

sid

es c

ongr

uen

t, a

ll

angl

es r

igh

t an

gles

).

F

ind

m∠

WZ

Y.

2. G

iven

: A

C =

AD

, an

d −

−

AB

⊥ −−

− B

D ,

m∠

DA

C =

44

and

−−−

CE

bis

ects

∠A

CD

.

Fin

d m

∠D

EC

. A

DC

BE

4. G

iven

: m

∠N

= 1

20,

−−

−

JN

� −

−−

MN

, �

JN

M �

�K

LM

.

Fin

d m

∠J

KM

.

J

K

L

MN

UV

W

XY

Z

4-6

15

78

148

45

037_

054_

GE

OC

RM

C04

_890

513.

indd

424/

12/0

812

:36:

12A

M

NA

ME

DA

TE

PE

RIO

D


Lesson 4-7

Cha

pte

r 4

43

G

lenc

oe G

eom

etry

Iden

tify

Co

ng

ruen

ce T

ran

sfo

rmat

ion

s A

con

gru

ence

tra

nsf

orm

atio

n i

s a

tran

sfor

mat

ion

wh

ere

the

orig

inal

fig

ure

, or

prei

mag

e, a

nd

the

tran

sfor

med

fig

ure

, or

imag

e, f

igu

re a

re s

till

con

gru

ent.

Th

e th

ree

type

s of

con

gru

ence

tra

nsf

orm

atio

ns

are

refl

ecti

on (

or f

lip)

, tra

nsl

atio

n (

or s

lide

), a

nd

rota

tion

(or

tu

rn).

Id

enti

fy t

he

typ

e of

con

gru

ence

tra

nsf

orm

atio

n s

how

n a

s a

refl

ecti

on, t

ran

sla

tion

, or

rota

tion

.

x

y

Ea

ch

ve

rte

x a

nd

its

im

ag

e a

re t

he

sa

me

dis

tan

ce

fro

m t

he

y-a

xis

.

Th

is is a

re

fle

ctio

n.

x

y

Ea

ch

ve

rte

x a

nd

its

im

ag

e a

re in

the

sa

me

po

sitio

n,

just

two

un

its

do

wn

. T

his

is a

tra

nsla

tio

n.

x

y

Ea

ch

ve

rte

x a

nd

its

im

ag

e a

re t

he

sa

me

dis

tan

ce

fro

m t

he

orig

in.

Th

e a

ng

les f

orm

ed

by e

ach

pa

ir o

f

co

rre

sp

on

din

g p

oin

ts a

nd

th

e

orig

in a

re c

on

gru

en

t. T

his

is a

rota

tio

n.

Exer

cise

sId

enti

fy t

he

typ

e of

con

gru

ence

tra

nsf

orm

atio

n s

how

n a

s a

refl

ecti

on, t

ran

sla

tion

, or

rot

ati

on.

1.

x

y

2.

x

y

3.

x

y

4.

x

y

5.

x

y

6.

x

y

Stud

y G

uide

and

Inte

rven

tion

Co

ng

ruen

ce T

ran

sfo

rmati

on

s

Exam

ple

4-7

refl

ecti

on

tran

sla

tio

nro

tati

on

rota

tio

nre

flecti

on

or

tran

sla

tio

nre

flecti

on

037_

054_

GE

OC

RM

C04

_890

513.

indd

434/

12/0

812

:36:

17A

M




An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass

NA

ME

DA

TE

PE

RIO

D


Cha

pte

r 4

44

G

lenc

oe G

eom

etry

Ver

ify

Co

ng

ruen

ce Y

ou c

an v

erif

y th

at r

efle

ctio

ns,

tra

nsl

atio

ns,

an

d ro

tati

ons

of

tria

ngl

es p

rodu

ce c

ongr

uen

t tr

ian

gles

usi

ng

SS

S.

V

erif

y co

ngr

uen

ce a

fter

a t

ran

sfor

mat

ion

.

�W

XY

wit

h v

erti

ces

W(3

, -7)

, X(6

, -7)

, Y(6

, -2)

is

a tr

ansf

orm

atio

n

of �

RS

T w

ith

ver

tice

s R

(2, 0

), S

(5, 0

), a

nd

T(5

, 5).

Gra

ph t

he

orig

inal

fig

ure

an

d it

s im

age.

Ide

nti

fy t

he

tran

sfor

mat

ion

an

d ve

rify

th

at i

t is

a c

ongr

uen

ce t

ran

sfor

mat

ion

.

Gra

ph e

ach

fig

ure

. Use

th

e D

ista

nce

For

mu

la t

o sh

ow t

he

side

s ar

e co

ngr

uen

t an

d th

e tr

ian

gles

are

con

gru

ent

by S

SS

.

RS

= 3

, ST

= 5

, TR

= √

��

��

��

�

( 5

- 2

) 2 + (

5 -

0) 2 =

√ �

�

34

WX

= 3

, XY

= 5

, YW

= √

��

��

��

��

�

( 6

- 3

) 2 + (

-2-

(-

7)) 2 =

√ �

�

34

−−

RS

� −−

− W

X ,

−−

ST

, � −

−

XY

, −

−

TR

� −−

− Y

W

By

SS

S, �

RS

T �

�W

XY

.

Exer

cise

s

CO

OR

DIN

ATE

GEO

MET

RY

Gra

ph

eac

h p

air

of t

rian

gles

wit

h t

he

give

n v

erti

ces.

T

hen

id

enti

fy t

he

tran

sfor

mat

ion

, an

d v

erif

y th

at i

t is

a c

ongr

uen

ce

tran

sfor

mat

ion

.

1. A

(−3,

1),

B(−

1, 1

), C

(−1,

4)

2. Q

(−3,

0),

R(−

2, 2

), S

(−1,

0)

D

(3, 1

), E

(1, 1

), F

(1, 4

) T

(2, −

4), U

(3, −

2), V

(4, −

4)

x

y

x

y

4-

7St

udy

Gui

de a

nd In

terv

enti

on (

con

tin

ued

)

Co

ng

ruen

ce T

ran

sfo

rmati

on

s

Exam

ple

x

y

refl

ecti

on

; A

B =

2,

BC

= 3

,

CA

=

√ �

�

13 ,

DE

= 2

, E

F =

3,

FD

= √

��

13 t

hu

s −

−

A

B �

−−

D

E ,

−−

B

C �

−−

E

F ,

−−

A

C �

−−

D

F B

y S

SS

,

�A

BC

�

�D

EF

tran

sla

tio

n;

QR

= √

�

5 ,

RS

= √

�

5 ,

SQ

= 2

, TU

= √

�

5 ,

UV

= √

�

5 ,

VT

= 2

th

us,

−−

Q

R �

−

−

TU

, −

−

R

S �

−

−

U

V ,

−−

S

Q �

−

−

V

T .

By S

SS

, �

QR

S �

�TU

V.

037_

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:36:

30A

M

NA

ME

DA

TE

PE

RIO

D


Lesson 4-7

Cha

pte

r 4

45

G

lenc

oe G

eom

etry

4-7

Iden

tify

th

e ty

pe

of c

ongr

uen

ce t

ran

sfor

mat

ion

sh

own

as

a re

flec

tion

, tra

nsl

ati

on,

or r

ota

tion

.

1.

x

y

2.

x

y

r

ota

tio

n

re

flecti

on

CO

OR

DIN

ATE

GEO

MET

RY

Id

enti

fy e

ach

tra

nsf

orm

atio

n a

nd

ver

ify

that

it

is a

co

ngr

uen

ce t

ran

sfor

mat

ion

.

3.

x

y

4.

x

y

T

ran

sla

tio

n;

AB

= √

�

5 ,

BC

= √

��

13 ,

CA

= 4

, D

E =

√ �

5 ,

EF =

√ �

�

13 ,

FD

= 4

. �

AB

C �

�D

EF b

y S

SS

.

R

efl

ecti

on

; Q

R =

5,

RS

= 4

,

SQ

= 3

, TU

= 5

, U

V =

4,

VT =

3.

�Q

RS

� �

TU

V b

y S

SS

.

CO

OR

DIN

ATE

GEO

MET

RY

Gra

ph

eac

h p

air

of t

rian

gles

wit

h t

he

give

n v

erti

ces.

T

hen

, id

enti

fy t

he

tran

sfor

mat

ion

, an

d v

erif

y th

at i

t is

a c

ongr

uen

ce

tran

sfor

mat

ion

.

5. A

(1, 3

), B

(1, 1

), C

(4, 1

);

6. J

(−3,

0),

K(−

2, 4

), L

(−1,

0);

D

(3, -

1), E

(1, -

1), F

(1, -

4)

Q(2

, −4)

, R(3

, 0),

S(4

, −4)

x

y

x

y

4-7

Skill

s Pr

acti

ceC

on

gru

en

ce T

ran

sfo

rmati

on

s

Tra

nsla

tio

n;

JK

= √

��

17 ,

KL =

√ �

�

17 ,

LJ =

2,

QR

= √

��

17 ,

RS

= √

��

17 ,

SQ

= 2

. �

JK

L �

�

QR

S b

y S

SS

.

rota

tio

n;

AB

= 2

, B

C =

3,

CA

=

√ �

�

13 ,

DE

= 2

, E

F =

3,

FD

= √

��

13

�A

BC

� �

DE

F b

y S

SS

.

037_

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2P

M




Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass

NA

ME

DA

TE

PE

RIO

D


Cha

pte

r 4

46

G

lenc

oe G

eom

etry

Iden

tify

th

e ty

pe

of c

ongr

uen

ce t

ran

sfor

mat

ion

sh

own

as

a re

flec

tion

, tra

nsl

ati

on,

or r

ota

tion

. 1

.

x

y

2.

x

y

3. I

den

tify

th

e ty

pe o

f co

ngr

uen

ce t

ran

sfor

mat

ion

sh

own

as

a re

flec

tion

, tra

nsl

atio

n, o

r ro

tati

on, a

nd

veri

fy t

hat

it

is a

con

gru

ence

tra

nsf

orm

atio

n.

R

ota

tio

n;

AB

= √

��

13 ,

BC

= 3

√ �

2 ,

CA

=

5

, A

D =

√ �

�

13

D

E =

3 √

�

2 ,

EA

= 5

, �

AB

C �

�A

DE

by S

SS

.

4. Δ

AB

C h

as v

erti

ces

A(−

4, 2

), B

(−2,

0),

C(−

4, -

2). Δ

DE

F h

as

vert

ices

D(4

, 2),

E(2

, 0),

F(4

, −2)

. Gra

ph t

he

orig

inal

fig

ure

an

d it

s im

age.

Th

en i

den

tify

th

e tr

ansf

orm

atio

n a

nd

veri

fy

that

it

is a

con

gru

ence

tra

nsf

orm

atio

n.

R

efl

ecti

on

; A

B =

2 √

�

2 ,

BC

= 2

√ �

2 ,

CA

= 4

, D

E =

2 √

�

2 ,

EF =

2 √

�

2 ,

FD

= 4

. �

AB

C �

�

DE

F b

y S

SS

.

5. S

TEN

CIL

S C

arly

is

plan

nin

g on

ste

nci

lin

g a

patt

ern

of

flow

ers

alon

g th

e ce

ilin

g in

her

be

droo

m. S

he

wan

ts a

ll o

f th

e fl

ower

s to

loo

k ex

actl

y th

e sa

me.

Wh

at t

ype

of c

ongr

uen

ce

tran

sfor

mat

ion

sh

ould

sh

e u

se?

Wh

y?

T

ran

sla

tio

n;

A t

ran

sla

tio

n c

on

tain

s c

on

gru

en

t sh

ap

es w

ith

th

e s

am

e

ori

en

tati

on

.

4-7

Prac

tice

Co

ng

ruen

ce T

ran

sfo

rmati

on

s

x

y

tran

sla

tio

nro

tati

on

x

y

037_

054_

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82:

47:0

1P

M

NA

ME

DA

TE

PE

RIO

D


Lesson 4-7

Cha

pte

r 4

47

G

lenc

oe G

eom

etry

4-7

Wor

d Pr

oble

m P

ract

ice

Co

ng

ruen

ce T

ran

sfo

rmati

on

s

1. Q

UIL

TIN

G Y

ou a

nd

you

r tw

o fr

ien

ds

visi

t a

craf

t fa

ir a

nd

not

ice

a qu

ilt,

par

t of

wh

ose

patt

ern

is

show

n b

elow

. You

r fi

rst

frie

nd

says

th

e pa

tter

n i

s re

peat

ed

by t

ran

slat

ion

. You

r se

con

d fr

ien

d sa

ys

the

patt

ern

is

repe

ated

by

rota

tion

. Wh

o is

cor

rect

?

B

oth

fri

en

ds a

re c

orr

ect.

Th

e

tria

ng

les a

re b

ein

g r

ota

ted

an

d

the h

exag

on

s a

re b

ein

g

tran

sla

ted

.

2. R

ECY

CLI

NG

Th

e in

tern

atio

nal

sym

bol

for

recy

clin

g is

sh

own

bel

ow. W

hat

typ

e of

con

gru

ence

tra

nsf

orm

atio

n d

oes

this

il

lust

rate

?

R

ota

tio

n

3. A

NA

TOM

Y C

arl

not

ices

th

at w

hen

he

hol

ds h

is h

ands

pal

m u

p in

fro

nt

of h

im,

they

loo

k th

e sa

me.

Wh

at t

ype

of

con

gru

ence

tra

nsf

orm

atio

n d

oes

this

il

lust

rate

?

R

efl

ecti

on

4. S

TAM

PS A

sh

eet

of p

osta

ge s

tam

ps

con

tain

s 20

sta

mps

in

4 r

ows

of 5

id

enti

cal

stam

ps e

ach

. Wh

at t

ype

of

con

gru

ence

tra

nsf

orm

atio

n d

oes

this

il

lust

rate

?

T

ran

sla

tio

n

5. M

OSA

ICS

José

cu

t re

ctan

gles

ou

t of

ti

ssu

e pa

per

to c

reat

e a

patt

ern

. He

cut

out

fou

r co

ngr

uen

t bl

ue

rect

angl

es a

nd

then

cu

t ou

t fo

ur

con

gru

ent

red

rect

angl

es t

hat

wer

e sl

igh

tly

smal

ler

to

crea

te t

he

patt

ern

sh

own

bel

ow.

a. W

hic

h c

ongr

uen

ce t

ran

sfor

mat

ion

do

es t

his

pat

tern

ill

ust

rate

?

Ro

tati

on

b.

If t

he

wh

ole

squ

are

wer

e ro

tate

d 90

, w

ould

th

e tr

ansf

orm

atio

n s

till

be

the

sam

e?

Yes

037_

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01A

M




An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass

NA

ME

DA

TE

PE

RIO

D


Cha

pte

r 4

48

G

lenc

oe G

eom

etry

4-7

Enri

chm

ent

Tra

nsfo

rmati

on

s i

n T

he C

oo

rdin

ate

Pla

ne

Th

e fo

llow

ing

stat

emen

t te

lls

one

way

to

rela

te p

oin

ts t

o

corr

espo

ndi

ng

tran

slat

ed p

oin

ts i

n t

he

coor

din

ate

plan

e.

(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

m

appe

d to

th

e po

int

wit

h c

oord

inat

es (

x +

6, y

- 3

).”

Wit

h t

his

tra

nsf

orm

atio

n, f

or e

xam

ple,

(3,

5)

is m

appe

d or

m

oved

to

(3 +

6, 5

- 3

) or

(9,

2).

Th

e fi

gure

sh

ows

how

th

e tr

ian

gle

AB

C i

s 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.

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)

3. (x

, y)

→ (

x +

5, y

+ 4

)

4. (

x, y

) →

(-

x , -

y)

5. (x

, y)

→ (

- 1 −

2 x,

y)

x

y

Ox

y

O

x

y

Ox

y

O

x

y

B

A

C

X

ZYO

(x,y

)� (x

+ 6

, y-

3)

yes

yes

yes

no

Yes;

the t

ran

sfo

rmati

on

slid

es t

he f

igu

re t

o t

he l

ow

er

rig

ht

wit

ho

ut

ch

an

gin

g i

ts s

ize o

r sh

ap

e.

037_

054_

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:37:

07A

M

NA

ME

DA

TE

PE

RIO

D


Lesson 4-8

Cha

pte

r 4

49

G

lenc

oe G

eom

etry

Stud

y G

uide

and

Inte

rven

tion

Tri

an

gle

s a

nd

Co

ord

inate

Pro

of

Posi

tio

n a

nd

Lab

el T

rian

gle

s A

coo

rdin

ate

proo

f u

ses

poin

ts, d

ista

nce

s, a

nd

slop

es t

o pr

ove

geom

etri

c pr

oper

ties

. Th

e fi

rst

step

in

wri

tin

g a

coor

din

ate

proo

f is

to

plac

e a

figu

re

on t

he

coor

din

ate

plan

e an

d la

bel

the

vert

ices

. Use

th

e fo

llow

ing

guid

elin

es.

1.

Use

th

e o

rig

in a

s a

ve

rte

x o

r ce

nte

r o

f th

e f

igu

re.

2.

Pla

ce

at

lea

st

on

e s

ide

of

the

po

lyg

on

on

an

axis

.

3.

Ke

ep

th

e f

igu

re in

th

e f

irst

qu

ad

ran

t if p

ossib

le.

4.

Use

co

ord

ina

tes t

ha

t m

ake

co

mp

uta

tio

ns a

s s

imp

le a

s p

ossib

le.

Pos

itio

n a

n i

sosc

eles

tri

angl

e on

th

e co

ord

inat

e p

lan

e so

th

at i

ts s

ides

are

a u

nit

s lo

ng

and

on

e si

de

is o

n t

he

pos

itiv

e x-

axis

.

Sta

rt w

ith

R(0

, 0).

If

RT

is

a, t

hen

an

oth

er v

erte

x is

T(a

, 0).

For

ver

tex

S, t

he

x-co

ordi

nat

e is

a −

2 .

Use

b f

or t

he

y-co

ordi

nat

e,

so t

he

vert

ex i

s S

( a −

2 ,

b ) .

Exer

cise

sN

ame

the

mis

sin

g co

ord

inat

es o

f ea

ch t

rian

gle.

1.

x

y

B( 2

p, 0

)

C( ?

,q)

A( 0

, 0)

2.

x

y

S( 2

a, 0

)

T(?,

?)

R( 0

, 0)

3.

x

y

G( 2

g, 0

)

F( ?

,b)

E( ?

, ?)

Pos

itio

n a

nd

lab

el e

ach

tri

angl

e on

th

e co

ord

inat

e p

lan

e.

4. i

sosc

eles

tri

angl

e

5. is

osce

les

righ

t �

DE

F

6. e

quil

ater

al t

rian

gle

�R

ST

wit

h b

ase

−−

RS

w

ith

leg

s e

un

its

lon

g �

EQ

I w

ith

ver

tex

4a u

nit

s lo

ng

Q

(0, √

�

3 b)

an

d si

des

2b

un

its

lon

g

x

y

I( b, 0

)E

( –b,

0)

Q( 0

,√3⎯b

)

x

y

E( e

, 0)

F( e

,e)

D( 0

, 0)

x

yT(

2a,b

)

R( 0

, 0)

S( 4

a, 0

)

x

y

T(a,

0)

R( 0

, 0)S(a – 2,b

)

Sam

ple

an

sw

ers

are

giv

en

.

C

(p,

q)

T(2

a,

2a

) E

(-2g

, 0);

F(0

, b

)

Exam

ple

4-8

037_

054_

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:59:

37A

M




Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass

NA

ME

DA

TE

PE

RIO

D


Cha

pte

r 4

50

G

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

to

veri

fy p

rope

rtie

s. M

any

coor

din

ate

proo

fs u

se t

he

Dis

tan

ce F

orm

ula

, Slo

pe F

orm

ula

, or

Mid

poin

t T

heo

rem

.

P

rove

th

at a

seg

men

t fr

om t

he

vert

ex

angl

e of

an

iso

scel

es t

rian

gle

to t

he

mid

poi

nt

of t

he

bas

e is

per

pen

dic

ula

r to

th

e b

ase.

Fir

st, p

osit

ion

an

d la

bel

an i

sosc

eles

tri

angl

e on

th

e co

ordi

nat

e pl

ane.

On

e w

ay i

s to

use

T(a

, 0),

R(-

a, 0

), a

nd

S(0

, c).

Th

en U

(0, 0

) is

th

e m

idpo

int

of −

−

RT

.

Giv

en: I

sosc

eles

�R

ST

; U i

s th

e m

idpo

int

of b

ase

−−

RT

.P

rove

: −−

− S

U ⊥

−−

RT

Pro

of:

U i

s th

e m

idpo

int

of −

−

RT

so

the

coor

din

ates

of

U a

re ( -

a +

a

−

2 ,

0 +

0

−

2 ) =

(0,

0).

Th

us

−−−

SU

lie

s on

the

y-ax

is, a

nd

�R

ST

was

pla

ced

so −

−

RT

lie

s on

th

e x-

axis

. Th

e ax

es a

re p

erpe

ndi

cula

r, s

o −−

− S

U ⊥

−−

RT

.

Exer

cise

PRO

OF

Wri

te a

coo

rdin

ate

pro

of f

or t

he

stat

emen

t.

Pro

ve t

hat

th

e se

gmen

ts jo

inin

g th

e m

idpo

ints

of

the

side

s of

a r

igh

t tr

ian

gle

form

a r

igh

t tr

ian

gle.

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)

Sam

ple

an

sw

er:

Po

sit

ion

an

d l

ab

el

rig

ht

�A

BC

wit

h t

he c

oo

rdin

ate

s

A(0

, 0),

B(0

, 2b

), a

nd

C(2

a,

0).

Th

e m

idp

oin

t P

of

BC

is

( 0

+ 2

a

−

2

, 2

b +

0

−

2

) =

(a

, b

).

Th

e m

idp

oin

t Q

of

AC

is (

0 +

2

a

−

2

, 2

+ 0

−

2

) =

(a,

0).

Th

e m

idp

oin

t R

of

AB

is

( 0

+ 0

−

2

, 0

+ 2

b

−

2

) =

(0,

b).

Th

e s

lop

e o

f −

−

R

P i

s b

- b

−

a -

0 =

0

−

a =

0,

so

th

e s

eg

men

t is

ho

rizo

nta

l.

Th

e s

lop

e o

f −

−

P

Q i

s

b -

0

−

a -

a =

b

−

0 ,

wh

ich

is u

nd

efi

ned

, so

th

e s

eg

men

t is

vert

ical.

∠R

PQ

is a

rig

ht

an

gle

becau

se a

ny h

ori

zo

nta

l lin

e i

s p

erp

en

dic

ula

r to

an

y

vert

ical

lin

e.

�P

RQ

has a

rig

ht

an

gle

, so

�P

RQ

is a

rig

ht

tria

ng

le.

4-8

Stud

y G

uide

and

Inte

rven

tion

(co

nti

nu

ed)

Tri

an

gle

s a

nd

Co

ord

inate

Pro

of

Exam

ple

037_

054_

GE

OC

RM

C04

_890

513.

indd

504/

12/0

812

:37:

18A

M

NA

ME

DA

TE

PE

RIO

D


Lesson 4-8

Cha

pte

r 4

51

G

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. r

igh

t �

FG

H w

ith

leg

s

2. is

osce

les

�K

LP

wit

h

3. is

osce

les

�A

ND

wit

ha

un

its

and

b u

nit

s lo

ng

bas

e −

−

KP

6b

un

its

lon

g b

ase

−−−

AD

5a

un

its

lon

g

x

y F( 0

,a)

G( 0

, 0)

H( b

, 0)

x

yL(

3b,c

)

K( 0

, 0)

P( 6

b, 0

)

x

yN

(5 – 2a,

b )

A( 0

, 0)

D( 5

a, 0

)

Nam

e th

e m

issi

ng

coor

din

ates

of

each

tri

angl

e. 4

.

x

y

B( 2

a, 0

)

A( 0

, ?)

C( 0

, 0)

5.

x

y

Y( 2

b, 0

)

Z( ?

, ?)

X( 0

, 0)

6.

x

y

N( 3

b, 0

)

M( ?

, ?)

O( 0

, 0)

7.

x

y

Q( ?

, ?)

R( 2

a,b)

P( 0

, 0)

8.

x

y

P( 7

b, 0

)

R( ?

, ?)

N( 0

, 0)

9.

x

y

U( a

, 0)

T(?,

?)

S( –

a, 0

)

10. P

RO

OF

Wri

te a

coo

rdin

ate

proo

f to

pro

ve t

hat

in

an

iso

scel

es r

igh

t tr

ian

gle,

th

e se

gmen

t fr

om t

he

vert

ex o

f th

e ri

ght

angl

e to

th

e m

idpo

int

of t

he

hyp

oten

use

is

perp

endi

cula

r to

th

e h

ypot

enu

se.

G

iven

: is

osce

les

righ

t �

AB

C w

ith

∠A

BC

th

e ri

ght

angl

e an

d M

th

e m

idpo

int

of −

−

AC

P

rove

: −

−−

BM

⊥ −

−

AC

x

y

C( 2

a, 0

)

A( 0

, 2a)

M

B( 0

, 0)

Sam

ple

an

sw

ers

are

giv

en

.

2

a

Z(b

, b

√ �

3 )

M(0

, c)

Q

(4a

, 0)

R (

7

−

2 b

, c

)

T(∅

, b

)

P

roo

f:

T

he M

idp

oin

t F

orm

ula

sh

ow

s t

hat

the c

oo

rdin

ate

s o

f

M

are

( 0

+ 2a

−

2

, 2

a +

0

−

2

) o

r (a

, a).

Th

e s

lop

e o

f −

−

A

C i

s

2a -

0

−

0 -

2

a

= -

1.

Th

e s

lop

e o

f −

−

B

M i

s

a -

0

−

a -

0

= 1

. T

he p

rod

uct

o

f th

e s

lop

es i

s -

1,

so

−−

B

M ⊥

−−

A

C .

4-8

Skill

s Pr

acti

ceTri

an

gle

s a

nd

Co

ord

inate

Pro

of

037_

054_

GE

OC

RM

C04

_890

513.

indd

515/

22/0

88:

07:5

2P

M




An

swer

s

Co

pyr

ight

© G

lenc

oe/

McG

raw

-Hill

, a d

ivis

ion

of

The

McG

raw

-Hill

Co

mp

anie

s, In

c.

Pdf Pass

NA

ME

DA

TE

PE

RIO

D


Cha

pte

r 4

52

G

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. e

quil

ater

al �

SW

Y w

ith

2.

isos

cele

s �

BL

P w

ith

3.

isos

cele

s ri

ght

�D

GJ

s

ides

1 −

4 a

un

its

lon

g b

ase

−−

BL

3b

un

its

lon

g w

ith

hyp

oten

use

−−

DJ

an

d le

gs 2

a u

nit

s lo

ng

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)

Nam

e th

e m

issi

ng

coor

din

ates

of

each

tri

angl

e.

4.

x

yS

( ?, ?

)

J(0,

0)

R(

b, 0

)1 – 3

5.

x

y

C( ?

, 0)

E( 0

, ?)

B( –

3a, 0

)

6.

x

y

P( 2

b, 0

)

M( 0

, ?)

N( ?

, 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. P

roo

f 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,

an

d th

e m

all

are

at t

he

vert

ices

of

a ri

ght

tria

ngl

e.

G

iven

: �

SK

M

P

rove

: �

SK

M i

s a

righ

t tr

ian

gle.

8. F

ind

the

dist

ance

bet

wee

n t

he

mal

l an

d K

arin

a’s

hom

e.

x

y S( 0

, 0)

K( 6

, 4)

M( –

2, 3

)

S

( 1

−

6 b

, c

)

C(3

a,

0),

E(0

, c)

M(0

, c

), N

(-2

b,

0)

P

roo

f:

S

lop

e o

f S

K =

4 -

0

−

6 -

0 o

r 2

−

3

S

lop

e o

f S

M =

3

- 0

−

-2

- 0 o

r -

3

−

2

S

ince t

he s

lop

e o

f −

−

S

M i

s t

he n

eg

ati

ve r

ecip

rocal

of

the

s

lop

e o

f −

−

S

K ,

−−

S

M ⊥

−−

S

K .

Th

ere

fore

, �

SK

M i

s r

igh

t tr

ian

gle

.

K

M =

√ �

��

��

��

��

(-

2 -

6) 2

+ (

3 -

4) 2

= √

��

�

64 +

1 =

√ �

�

65 o

r ≈

8.1

miles

4-8

Prac

tice

Tri

an

gle

s a

nd

Co

ord

inate

Pro

of

Sam

ple

an

sw

ers

are

g

iven

.

037_

054_

GE

OC

RM

C04

_890

513.

indd

524/

12/0

812

:37:

29A

M

NA

ME

DA

TE

PE

RIO

D


Lesson 4-8

Cha

pte

r 4

53

G

lenc

oe G

eom

etry

1. S

HEL

VES

Mar

tha

has

a s

hel

f br

acke

t sh

aped

lik

e a

righ

t is

osce

les

tria

ngl

e.

Sh

e w

ants

to

know

th

e le

ngt

h o

f th

e h

ypot

enu

se r

elat

ive

to t

he

side

s. S

he

does

not

hav

e a

rule

r, b

ut

rem

embe

rs

the

Dis

tan

ce F

orm

ula

. Sh

e pl

aces

th

e br

acke

t on

a c

oord

inat

e gr

id w

ith

th

e ri

ght

angl

e at

th

e or

igin

. Th

e le

ngt

h o

f ea

ch l

eg i

s a.

Wh

at a

re t

he

coor

din

ates

of

th

e ve

rtic

es m

akin

g th

e tw

o ac

ute

an

gles

?(a

, 0)

an

d (

0,

a)

2. F

LAG

S A

fla

g is

sh

aped

lik

e an

iso

scel

es

tria

ngl

e. A

des

ign

er w

ould

li

ke t

o m

ake

a dr

awin

g of

th

e fl

ag o

n a

coo

rdin

ate

plan

e.

Sh

e po

siti

ons

it s

o th

at t

he

base

of

the

tria

ngl

e is

on

th

e y-

axis

wit

h o

ne

endp

oin

t lo

cate

d at

(0,

0).

Sh

e lo

cate

sth

e ti

p of

th

e fl

ag a

t (a

, b − 2

).W

hat

are

th

e co

ordi

nat

es o

f th

e th

ird

vert

ex?

(0,

b)

3. B

ILLI

AR

DS

Th

e fi

gure

sh

ows

a si

tuat

ion

on

a b

illi

ard

tabl

e.(–

b, ?

)y

xO

( 0, 0

)

Wh

at a

re t

he

coor

din

ates

of

the

cue

ball

be

fore

it

is h

it a

nd

the

poin

t w

her

e th

e cu

e ba

ll h

its

the

edge

of

the

tabl

e?

Sam

ple

an

sw

er:

(-

2b

, 0),

(-

b,

a)

4. T

ENTS

Th

e en

tran

ce t

o M

att’s

ten

t m

akes

an

iso

scel

es t

rian

gle.

If

plac

ed o

n

a co

ordi

nat

e gr

id w

ith

th

e ba

se o

n t

he

x–ax

is a

nd

the

left

cor

ner

at

the

orig

in,

the

righ

t co

rner

wou

ld b

e at

(6,

0)

and

the

vert

ex a

ngl

e w

ould

be

at (

3, 4

).

Pro

ve t

hat

it

is a

n i

sosc

eles

tri

angl

e.

Sam

ple

an

sw

er:

Bo

th s

eg

men

ts

of

the t

rian

gle

fro

m t

he b

ase t

o

the v

ert

ex a

re 5

un

its l

on

g.

Sin

ce

the t

wo

sid

es a

re c

on

gru

en

t, i

t is

an

iso

scele

s t

rian

gle

.

5. D

RA

FTIN

GA

n e

ngi

nee

r is

des

ign

ing

a ro

adw

ay. T

hre

e ro

ads

inte

rsec

t to

fo

rm a

tri

angl

e. T

he

engi

nee

r m

arks

tw

o po

ints

of

the

tria

ngl

e at

(-

5, 0

) an

d (5

, 0)

on a

coo

rdin

ate

plan

e.

a. D

escr

ibe

the

set

of p

oin

ts i

n t

he

coor

din

ate

plan

e th

at c

ould

not

be

use

d as

th

e th

ird

vert

ex o

f th

e tr

ian

gle.

the x

-axis

b.

Des

crib

e th

e se

t of

poi

nts

in

th

e co

ordi

nat

e pl

ane

that

wou

ld m

ake

the

vert

ex o

f an

iso

scel

es t

rian

gle

toge

ther

wit

h t

he

two

con

gru

ent

side

s.

the y

-axis

excep

t fo

r th

e o

rig

in

c. D

escr

ibe

the

set

of p

oin

ts i

n t

he

coor

din

ate

plan

e th

at w

ould

mak

e a

righ

t tr

ian

gle

wit

h t

he

oth

er t

wo

poin

ts i

f th

e ri

ght

angl

e is

loc

ated

at

(-5,

0).

the p

oin

ts w

ith

x-c

oo

rdin

ate

-

5,

excep

t fo

r (-

5,

0)

Wor

d Pr

oble

m P

ract

ice

Tri

an

gle

s a

nd

Co

ord

inate

Pro

of

4-8

037_

054_

GE

OC

RM

C04

_890

513.

indd

534/

12/0

812

:37:

37A

M




Co

pyrig

ht © G

lencoe/M

cGraw

-Hill, a d

ivision o

f The M

cGraw

-Hill C

om

panies, Inc.

Pdf Pass

NA

ME

DA

TE

PE

RIO

D


Cha

pte

r 4

54

G

lenc

oe G

eom

etry

Recta

ng

le P

ara

do

xU

se g

rid

pape

r to

dra

w t

he

two

rect

angl

es b

elow

. Cu

t th

em i

nto

pie

ces

alon

g th

e da

shed

lin

es. T

hen

rea

rran

ge t

he

piec

es t

o fo

rm a

new

rec

tan

gle.

1. M

ake

a sk

etch

of

the

new

rec

tan

gle.

Lab

el t

he

wh

ole

nu

mbe

r le

ngt

hs

2. W

hat

is

the

com

bin

ed a

rea

of t

he

two

orig

inal

rec

tan

gles

?

3

9 u

nit

s2

3. W

hat

is

the

area

of

the

new

rec

tan

gle?

4

0 u

nit

s2

4. M

ake

a co

nje

ctu

re a

s to

wh

y th

ere

is a

dis

crep

ancy

bet

wee

n t

he

answ

ers

to

Exe

rcis

es 2

an

d 3.

S

ee s

tud

en

ts’

an

sw

ers

.

5. U

se a

str

aigh

t ed

ge t

o pr

ecis

ely

draw

th

e fo

ur

shap

es t

hat

m

ake

up

the

larg

er r

ecta

ngl

e. W

hat

doe

s yo

u d

raw

ing

show

?

I

t sh

ow

s a

gap

. S

o,

the r

ecta

ng

les d

on

’t q

uit

e

co

me t

og

eth

er

like t

hey l

oo

k l

ike t

hey w

ill.

6. H

ow c

ould

you

use

slo

pe t

o sh

ow t

hat

you

dra

win

g is

acc

ura

te?

T

he s

lop

e o

f th

e b

ott

om

ed

ge o

f th

e t

op

left

pie

ce a

nd

th

e s

lop

e o

f th

e

bo

tto

m e

dg

e o

f th

e a

dja

cen

t p

iece a

re n

ot

the s

am

e s

o t

he t

wo

pie

ces d

o

no

t fi

t to

geth

er

to m

ake a

str

aig

ht

lin

e a

s t

he m

od

el

su

gg

ests

th

ey w

ou

ld.

53

5

5

3

3

3

3

5

3

35

5

4-8

Enri

chm

ent

037_

054_

GE

OC

RM

C04

_890

513.

indd

544/

12/0

812

:37:

42A

M




Copyright

© G

lencoe/M

cG

raw

-Hill

, a d

ivis

ion o

f T

he M

cG

raw

-Hill

Com

panie

s,

Inc.

An

swer

s


PDF Pass

Chapter 4 Assessment Answer KeyQuiz 1 (Lessons 4-1 and 4-2) Quiz 3 (Lessons 4-5 and 4-6) Mid-Chapter TestPage 57 Page 58 Page 59

Quiz 2 (Lessons 4-3 and 4-4) Quiz 4 (Lessons 4-7 and 4-8)

Page 57 Page 58

1.

2.

3.

4.

5. 6. 7. 8. 9. 10.

obtuse scalene

B

x = 2, AB = 15,

BC = 15, AC = 10

AB = √ ��

53 , BC = 2 √ ��

10 ,

AC = √ ��

41 ; scalene

37

42

132

73

73

30

1.

2.

3.

4.

5.

�KMN � �KML

∠O

x = 8; y = 14

SSS

∠Q � ∠S

1.

2.

3.

4.

5.

Def. of angle bisector

AAS

60

45

120

1.

2.

3.

4.

5.

translation

x

y

J(a–2, 0)

D(0, a)

L(0, 0)

(a √ � 3 , 0)

AC = √ ��

34 , AB = 6,

and CB = √ ��

34

−−

AC � −−

CB

1.

2.

3.

4.

D

H

A

G

5.

6.

7.

8.

AB = √ ��

41 ,

BC = √ ��

29 ,

AC = 5 √ �

2 ; scalene

SAS

3

congruent

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Chapter 4 Assessment Answer KeyVocabulary Test Form 1 Page 60 Page 61 Page 62

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11.

12.

false, equiangular triangle

true

congruence transformation

exterior

included angle

flow proof

right triangle

included side

coordinate proof

vertex angle

a statement that can easily be

proved using a theorem

two triangles in which all

corresponding parts are

congruent

1.

2.

3.

4.

5.

6.

7.

8.

C

J

A

H

B

J

C

G

9.

10.

11.

12.

13.

C

F

C

F

B

isoscelesB:

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Chapter 4 Assessment Answer KeyForm 2A Form 2B Page 63 Page 64 Page 65 Page 66

9.

10.

11.

12.

13.

B

H

B

F

A

B: A(0, 0), C(-a, 0)

1.

2.

3.

4.

5.

6.

7.

8.

D

G

D

F

C

G

D

J

9.

10.

11.

12.

13.

C

H

D

F

D

B: -2

1.

2.

3.

4.

5.

6.

7.

8.

C

H

A

J

B

J

D

F


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Chapter 4 Assessment Answer KeyForm 2C Page 67 Page 68

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

reflection;

AB = √ �

29 , BC = √ �

10 ,

CA = √ ��

17 , QT = √ �

29 ,

ST = √ �

10 , QS = √ �

17 ,

9.

10.

11.

12.

13.

14.

B:

45, 45, 90

Isosceles �

Theorem, AAS

50

x = 4

P ( b − 2 , b −

2 ) , AB = 2CP

�XYZ � �MNO

x

yC(b–

2, c)

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


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Chapter 4 Assessment Answer KeyForm 2D Page 69 Page 70

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

45, 45, 90

rotation; JK = √ ��

10 ,

KL = √ ��

37 , LJ = √ ��

29 ,

WX = √ ��

10 , XY =

√ �� 37 ,

YW = √ �

29

10.

11.

12.

13.

14.

B:

Isosceles � Theorem, AAS

50

x = 6

M ( a −

2 , 0) , N (0,

b −

2 )

�ABC � �DEF

x

y

M(3a, 0)

L(1.5a, b)

K(0, 0)

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Chapter 4 Assessment Answer KeyForm 3 Page 71 Page 72

1.

2.

3.

4.

5.

6.

7.

8.

x = 4,

AB = BC = 78, AC = 100

AB = 5, BC = 10, AC = 3 √ � 5 ; scalene

obtuse

20

90

40

reflection

GH = JK = 2 √ �

5 , IG = LJ

= 2 √ �

5 , IH = LK = 2 √ �

2 ;

�GHI � �JKL by SSS.

AAS

9.

10.

11.

12.

13.

B:

Isosceles

Triangle Theorem

SAS

x = 3

Hypotenuse-leg

congruence for

right �s

m∠1 = m∠3 =

m∠4 = m∠6 = m∠9 = 20,

m∠2 = m∠5 = 40, m∠8 = 60, m∠7 = 140

x

y

B(a + b, 0)

C(a + b, c)2

A(0, 0)


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Chapter 4 Assessment Answer KeyExtended-Response Test, Page 73

Scoring Rubric

Score General Description Specific Criteria

4 Superior

A correct solution that

is supported by well-

developed, accurate

explanations

• Shows thorough understanding of the concepts of using the Distance Formula to classify triangles and verify congruence, finding missing angles, solving algebraic equations in isosceles and equilateral triangles, proving triangles congruent, congruence 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.

3 Satisfactory

A generally correct solution,

but may contain minor flaws

in reasoning or computation

• Shows understanding of the concepts of using the Distance Formula to classify triangles and verify congruence, finding missing angles, solving algebraic equations in isosceles and equilateral triangles, proving triangles congruent, congruence and writing 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.

2 Nearly Satisfactory

A partially correct

interpretation and/or

solution to the problem

• Shows understanding of most of the concepts of using the Distance Formula to classify triangles and verify congruence, finding missing angles, solving algebraic equations in isosceles and equilateral triangles, proving triangles congruent, congrueence and writing 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.

1 Nearly Unsatisfactory

A correct solution with no

supporting evidence or

explanation

• 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.

0 Unsatisfactory

An incorrect solution

indicating no mathematical

understanding of the concept

or task, or no solution is given

• Shows little or no understanding of most of the concepts

of using the Distance Formula to classify triangles and verify congruence, finding missing angles, solving algebraic equations in isosceles and equilateral triangles, proving triangles congruent, congruence and writing 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.


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Chapter 4 Assessment Answer KeyExtended-Response Test, Page 73

Sample Answers

1. a. The figure is an acute isosceles triangle. b. 9x + 4 + 2(20x - 10) = 180 9x + 4 + 40x - 20 = 180 49x - 16 = 180 49x = 196 x = 4

2. a. To determine whether a triangle is equilateral, the coordinates should be graphed first to see if they form a triangle. Then list the segments which form the triangle and use the Distance Formula to find their lengths. A triangle is equilateral only if all three sides have equal measures.

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, m∠1 = 180 - 78 - 58 or 44. Lastly, since ∠2 is an exterior angle, its measure is equivalent to the sum of the measures of the two remote interior angles, 58 + 78, which equals 136.

4. a. SSS postulate

b. ∠J � ∠G, ∠D � ∠E, ∠L � ∠S

−−−

DJ � −−−

EG , −−−

DL � −−−

ES , −−

JL � −−−

GS

5. Statements Reasons

1. −−−

AB || −−−

DE 1. Given

2. ∠ABC � ∠DEC 2. Alt. int. � are �.

3. −−−

AD bisects −−−

BE . 3. Given

4. −−−

BC � −−−

EC 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 A33, the following sample answers

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

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Chapter 4 Assessment Answer KeyStandardized Test Practice Page 74 Page 75

1.

2. F G H J

3. A B C D

4. F G H J

5.

6. F G H J

7. A B C D

A B C D

A B C D

8.

9.

10.

11.

12.

13.

14. F G H J

A B C D

F G H J

A B C D

F G H J

A B C D

F G H J

15.

16. 9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

6

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

9

8

7

6

5

4

3

2

1

0

1 8


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Chapter 4 Assessment Answer KeyStandardized Test Practice (continued) Page 76

17.

18.

19.

20.

21.

22a.

22b.

22c.

29 ft

35

62

40

∠P � ∠H, ∠Q � ∠G,

∠R � ∠B, −−

PQ � −−

HG ,

−−

QR � −−

GB , −−

PR � −−

HB

−−

FD

right triangle

15


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