Congruence - Oak Park Geometry Classes - Homeophsgc.weebly.com/uploads/1/3/2/9/13299817/geometry... · · 2013-04-26SSS, SAS, ASA, and AAS | p. 319 6.3 p. 343Right Triangle Congruence

C H A P T E R

6 Congruence

6.1 Constructing Congruent Triangles or NotConstructing Triangles | p. 311

6.2 Congruence TheoremsSSS, SAS, ASA, and AAS | p. 319

6.3 Right Triangle Congruence TheoremsHL, LL, HA, and LA | p. 329

6.4 CPCTCCorresponding Parts of Congruent

Triangles are Congruent | p. 337

6.5 Isosceles Triangle TheoremsIsosceles Triangle Base Theorem,

Vertex Angle Theorem, Perpendicular

Bisector Theorem, Altitude to

Congruent Sides Theorem, and

Angle Bisector to Congruent Sides

Theorem | p. 343

6.6 Direct Proof vs. Indirect ProofInverse, Contrapositive, Direct Proof,

and Indirect Proof | p. 349

Chapter 6 | Congruence 307

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Victorian houses in San Francisco, known as “Painted Ladies” for their bright exterior

colors, often share basic designs and floor plans. As a result, while the houses in this

image are decorated differently, their third floor dormers form congruent triangles,

triangles that have the same shape and the same size. You will explore the properties

of congruent triangles and use them as the basis for constructing mathematical proofs.

308 Chapter 6 | Congruence

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Introductory Problem for Chapter 6

Is Congruence a Special Case of Similarity or is Similarity a Special Case of Congruence?

Similar triangles are triangles that have the same shape.

A C

B

1. Construct �DEF so that �ABC � �DEF.

2. Describe the steps taken to construct �DEF.

3. Describe the relationship between the corresponding angles of the

similar triangles.

4. Describe the relationship between the corresponding sides of the

similar triangles.

Congruent triangles are triangles that have the same shape and the same size.

A

B

C

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Chapter 6 | Introductory Problem for Chapter 6 309

5. Construct �GHI so that �ABC � �GHI.

6. Describe the steps taken to construct �GHI.

7. Describe the relationship between the corresponding angles of the

congruent triangles.

8. Describe the relationship between the corresponding sides of the

congruent triangles.

9. Cessia says that all similar triangles are congruent. Ricky says that all

congruent triangles are similar. Who is correct? Explain.

Be prepared to share your solutions and methods.

6


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Lesson 6.1 | Constructing Congruent Triangles or Not 311

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Two polygons are congruent if they are the same shape and size.

1. Construct �ABC using the two line segments shown. Write the steps.

A

A C

B

PROBLEM 1 Construction

A Triangle Given Two Line Segments

Constructing Congruent Triangles or NotConstructing Triangles

6.1

OBJECTIVEIn this lesson you will:l Construct triangles to determine uniqueness.

6


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2. Classify �ABC based on the angles and the sides.

3. Compare the triangle that you constructed with the triangles that your

classmates constructed. What do you observe? Why?

4. Name the included angle for sides AB and AC.

5. Measure the included angle for sides AB and AC. Compare the measure of

your included angle with the measures of the included angles of your

classmates. What do you observe?

6. How does the measure of the included angle affect the length of side BC

of �ABC?

7. How many different triangles can be determined given the length of two sides

of a triangle?


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A Triangle Given Three Line Segments 1. Construct �ABC using the three line segments shown. Write the steps.

A

B C

A C

B




4. How many different triangles can be determined given the length of three sides

of a triangle?


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6 1. Construct �ABC using the two line segments and included angle shown.

Write the steps.

A

A

A

C

B


3. Compare the triangle that you constructed with the triangles that your class-

mates constructed. What do you observe? Why?

4. Could everyone construct an identical triangle if they were given �C or �B, the

angles that are not included? Explain.


A Triangle Given Two Line Segments and the Included Angle


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5. How many different triangles can be determined given:

a. The length of two sides of a triangle and the included angle?

b. The length of two sides of a triangle and the angle not included?


A Triangle Given Three Angles 1. Construct �ABC using the three angles shown. Write the steps.

AB

C


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2. Compare the triangle that you constructed with the triangles that your class-

mates constructed. What do you observe? Why?

3. Could everyone construct an identical triangle if they were given only two angles

of a triangle? Explain.

4. How many different triangles can be determined given three interior angles of

a triangle?


A Triangle Given Two Angles and One Line Segment 1. Construct a triangle using the two angles and the line segment shown.

Write the steps.


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3. How many different triangles can be determined given the measure of two

angles of a triangle and the length of one side? Explain your reasoning.

1. List all combinations of givens that determine a unique triangle.

2. List all combinations of givens that determine multiple triangles.

3. Did you use inductive or deductive reasoning to answer Questions 1 and 2?


PROBLEM 6 Summary


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Lesson 6.2 | Congruence Theorems 319

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6OBJECTIVESIn this lesson you will:l Identify congruent parts of triangles by SSS,

SAS, ASA, and AAS.l List the corresponding congruent parts of two

triangles from a congruence statement.l Distinguish between the Similarity Postulates

and the Congruence Theorems.

KEY TERMSl Side-Side-Side (SSS)

Congruence Theoreml Side-Angle-Side (SAS)

Congruence Theoreml Angle-Side-Angle (ASA)

Congruence Theoreml Angle-Angle-Side (AAS)

Congruence Theorem

6.2

PROBLEM 1 Making ConjecturesYou have identified various cases when a unique triangle can be constructed using

given sides or angles. This demonstrates that there is a congruent relationship between

two constructed triangles.

1. Write a conjecture for each congruent triangle relationship.

a. Given three sides.

b. Given two sides and the included angle.

c. Given two angles and one specified side.

These conjectures provide the basis for four theorems.

Congruence TheoremsSSS, SAS, ASA, and AAS

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The Side-Side-Side (SSS) Congruence Theorem states: “If three sides of one

triangle are congruent to the corresponding sides of another triangle, then the

triangles are congruent.”

1. Complete the two-column proof of the SSS Congruence Theorem.

B

AC

E

DF

Given: ___

AC � ___

DF , ___

AB � ___

DE , ___

BC � ___

EF

Prove: �ABC � �DEF

Statements Reasons

1. ___

AC � ___

DF , ____

AB � ____

DE , BC � ___

EF 1.

2. AC � , � DE, BC � 2. Definition of Congruence

3. AC ____ DF

� AB ____ DE

� BC ____ EF

� 3. Division Property of Equality

4. ____ DF

� AB ____ � BC ____ 4. Transitive Property of Equality

5. �ABC � �DEF 5.

6. �A � , ��E, �C � 6. Definition of similar triangles

7. �ABC � �DEF 7. Definition of congruent triangles

2. What is the difference between the SSS Similarity Postulate and the SSS

Congruence Theorem?

PROBLEM 2 Congruence Theorems


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The Side-Angle-Side (SAS) Congruence Theorem states: “If two sides and the

included angle of one triangle are congruent to the corresponding two sides and the

included angle of a second triangle, then the two triangles are congruent.”

3. Create a two-column proof of the SAS Congruence Theorem.

A C

B E

D F

Given: ___

AB � ___

DE , ___

AC � ___

DF , and �A � �D


Statements Reasons

4. What is the difference between the SAS Similarity Postulate and the SAS

Congruence Theorem?


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The Angle-Side-Angle (ASA) Congruence Theorem states: “If two angles and the

included side of one triangle are congruent to the corresponding two angles and the

included side of another triangle, the triangles are congruent.”

You will complete the proof of the ASA Congruence Theorem in the assignments for

this lesson.

5. Mark the appropriate sides and/or angles to prove �RWX � �CMT by ASA.

X T

R C

W M

6. Draw �ABC � �ABD by ASA and include appropriate markers.

The Angle-Angle-Side (AAS) Congruence Theorem states: “If two angles and the

non-included side of one triangle are congruent to the corresponding two angles and

the non-included side of another triangle, the triangles are congruent.”

You will complete the proof of the AAS Congruence Theorem in the assignments for

this lesson.

7. Mark the appropriate sides and/or angles to prove �RWX � �CMT by AAS.

X T

R C

W M


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8. Draw �ABC � �ABD by AAS and include appropriate markers.

9. Ricardo said the AAS method for proving two triangles congruent is really the

ASA method in disguise. Is Ricardo correct? Explain.

10. Is SAA a method for proving two triangles congruent? Explain.

11. What is the ratio of corresponding sides for two congruent triangles? Explain.


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1. Draw �JRB � �EMS. List the six pairs of congruent corresponding parts.

2. List the six pair of congruent corresponding parts if �GNP � �WCA.

3. Based on each hypothesis, is there enough information to conclude the

triangles shown are congruent? Explain your reasoning. Name the Congruence

Theorem, if applicable.

a. If ___

AB � ___

CD and ___

AE � ___

CE , is there enough information to determine whether

�ABE � �CDE?

A

E D

C

B

PROBLEM 3 Applying the Congruence Theorems


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b. If ___

RF � ___

BP and ___

BF � ___

RP , is there enough information determine whether

�RFP � �BPF?

R B

T

F P

c. If ____

WN � ___

HK , is there enough information to determine whether

�WNZ � �HKZ?

W

N

Z

K

H

d. If ___

JA � ____

MY and ____

YM bisects �JYA, is there enough information to determine

whether �JYM � �AYM?

J M

Y

A


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e. If ___

ST � ___

AR and ___

RS � ___

TA , is there enough information to determine whether

�STR � �ART?

S T

R A

f. If ____

GU bisects �BGD and ___

GB � ____

GD , is there enough information to determine

whether �GUD � �GUB?

G

D

B

U

g. If �CKM � �EKV, ___

CK � ___

EK , and ___

KV � ____

KM , is there enough information to

determine whether �KCV � �KEM?

M

V

E

C

K


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h. If �X � �Y and ___

BX � ___

BY , is there enough information to determine whether

�BXE � �BYT?

X

D

E YB

T

4. Draw two congruent triangles that share a common side and write the

congruence statement.

5. Draw two congruent triangles that share a common angle and write the

congruence statement.


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6. Six teams of students have the task of locating a specific point on an outdoor

basketball court. Determine whether each team can locate the point and

explain your reasoning.

a. The first team was given the distance between the posts for the basketball

hoops and the distance from each post to the point.

b. The second team was given the distance between the posts for the

basketball hoops and the measure of the angles from each post to the point.

c. The third team was given the distance between the posts for the basketball

hoops, the distance from one post to the point, and the measure of the

angle from the other post to the point.

d. The fourth team was given the distance between the posts for the basketball

hoops, the distance from one post to the point, and the measure of the

angle from that post to the point.

e. The fifth team was given the distance between the posts for the basketball

hoops, the measure of the angle from one post to the point, and the

measure of the angle from that point to the posts.

f. The sixth team was given the measures of all the angles.


Lesson 6.3 | Right Triangle Congruence Theorems 329

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1. List all of the Triangle Congruence Theorems.

2. How many pairs of corresponding parts are used in each congruence theorem?

The Congruence Theorems apply to all triangles. There are also theorems that only

apply to right triangles. Methods for proving two right triangles congruent are

somewhat shorter. You can prove two right triangles congruent using only two pairs

of corresponding parts.

3. Explain why only two pairs of corresponding parts are necessary to prove

two right triangles are congruent. What is special about right triangles that will

shorten the steps to prove two are congruent?

Let’s explore these methods.

OBJECTIVESIn this lesson you will:l Use given information to show two

right triangles are congruent.

l Prove the HL Congruence Theorem.

l Prove the LL Congruence Theorem.

l Prove the HA Congruence Theorem.

l Prove the LA Congruence Theorem.

KEY TERMSl Hypotenuse-Leg (HL) Congruence

Theoreml Leg-Leg (LL) Congruence Theoreml Hypotenuse-Angle (HA) Congruence

Theoreml Leg-Angle (LA) Congruence Theorem

6.3

PROBLEM 1 Right Triangle Congruence Theorems

Right Triangle Congruence TheoremsHL, LL, HA, and LA

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The Hypotenuse-Leg (HL) Congruence Theorem states: “If the hypotenuse and

leg of one right triangle are congruent to the hypotenuse and leg of another right

triangle, then the triangles are congruent.”

4. Create a two-column proof of the HL Congruence Theorem

A F E

C B D

Given: �C and �F are right angles

___

AC � ___

DF

___

AB � ___

DE


Statements Reasons


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The Leg-Leg (LL) Congruence Theorem states: “If two legs of one right triangle are

congruent to two legs of another right triangle, then the triangles are congruent.”

5. Create a two-column proof of the LL Congruence Theorem.

A F E

C B D


___

AC � ___

DF

___

CB � ___

FE


Statements Reasons

The Hypotenuse-Angle (HA) Congruence Theorem states: “If the hypotenuse and

an acute angle of one right triangle are congruent to the hypotenuse and acute

angle of another right triangle, then the triangles are congruent.”

6. Create a two-column proof of the HA Congruence Theorem.

A F E

C B D


___

AB � ___

DE

�A � �D


Statements Reasons


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The Leg-Angle (LA) Congruence Theorem states: “If a leg and an acute angle of

one right triangle are congruent to a leg and an acute angle of another right triangle,

then the triangles are congruent.”

7. Create a two-column proof of the LA Congruence Theorem.

A F E

C B D


___

AC � ___

DF

�A � �D


Statements Reasons


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Determine if there is enough information to prove the two triangles are congruent.

If so, name the Congruence Theorem used.

1. If ___

CS � ___

SD , ____

WD � ___

SD , and P is the midpoint of ____

CW , is �CSP � �WDP?

C

S

P D

W

2. If ___

RF � ___

FP , ___

BP � ___

FP , and ___

RP and ___

FB bisect each other, is �RFP � �BPF?

F P

BR

T

PROBLEM 2 Applying Right Triangle Congruence Theorems


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3. Pat always trips on the third step and she thinks that step may be a different

size. The contractor told her that all the treads and risers are perpendicular

to each other. Is that enough information to state that the steps are the same

size? In other words, if ____

WN � ___

NZ and ___

ZH � ___

HK , is �WNZ � �ZHK?

4. If ___

JA � ____

MY and ___

JY � ___

AY , is �JYM � �AYM?

J M

Y

A

5. If ___

ST � ___

SR , ___

AT � ___

AR , and �STR � �ATR, is �STR � �ATR?

S

R

A

T


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It is necessary to make a statement about the presence of right triangles when you

use the Right Triangle Congruence Theorems. If you have previously identified them,

the reason is the definition of right angles.

6. Create a two-column proof of the following.

Given: ____

GU � ___

DB

D

UG

B

___

GB � ____

GD

Prove: �GUD � �GUB

Statements Reasons

7. Create a two-column proof of the following.

Given: ____

GU is the � bisector of ___

DB

D

UG

B

Prove: �GUD � �GUB

Statements Reasons


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8. A friend wants to place a post in a lake 20 feet straight out from the dock.

What is the minimum information you need to make sure the angle formed

by the edge of the dock and the post is a right angle?


Lesson 6.4 | CPCTC 337

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6OBJECTIVESIn this lesson you will:l Identify corresponding parts of

congruent triangles.l Use corresponding parts of congruent triangles

are congruent to prove angles and segments are congruent.

l Use corresponding parts of congruent triangles are congruent to prove the Isosceles Triangle Base Angle Theorem.

l Use corresponding parts of congruent triangles are congruent to prove the Isosceles Triangle Base Angle Converse Theorem.

KEY TERMSl corresponding parts of congruent

triangles are congruent (CPCTC)l Isosceles Triangle Base

Angle Theoreml Isosceles Triangle Base

Angle Converse Theorem

6.4 CPCTCCorresponding Parts of Congruent Triangles are Congruent

PROBLEM 1 CPCTCIf two triangles are congruent, then each part of one triangle is congruent to the

corresponding part of the other triangle. “Corresponding parts of congruent triangles are congruent,” abbreviated as CPCTC, is often used for a reason in

proof problems. CPCTC states that corresponding angles or sides in two congruent

triangles are congruent. This reason can only be used after you have proven that the

triangles are congruent.

To use CPCTC in a proof, follow these steps:

Step 1: Identify two triangles in which segments or angles are

corresponding parts.

Step 2: Prove the triangles congruent.

Step 3: State the two parts are congruent using CPCTC as the reason.

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1. Create a two-column proof.

Given: ____

CW and ___

SD bisect each other

Prove: ___

CS � ____

WD

Statements Reasons


Given: ___

SU � ___

SK , ___

SR � ___

SH

Prove: �U � �K

Statements Reasons

C

S

P D

W

S H K

R

U


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CPCTC makes it possible to prove other theorems. For example:

The Isosceles Triangle Base Angle Theorem states: “If two sides of a triangle are

congruent, then the angles opposite these sides are congruent.”

To prove the Isosceles Triangle Base Angle Theorem, you need to add an auxiliary

line

to an isosceles triangle that bisects the vertex angle as shown.


Given: ___

GB � ____

GD

Prove: �B � �D

D

UG

B

Statements Reasons

PROBLEM 2 Isosceles Triangle Base Angle Theorem and Its Converse


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The Isosceles Triangle Base Angle Converse Theorem states: “If two angles of a

triangle are congruent, then the sides opposite these angles are congruent.”

To prove the Isosceles Triangle Base Angle Converse Theorem, you need to add an

auxiliary line to an isosceles triangle that bisects the vertex angle as shown.

2. Create a two-column proof. D

UG

B

Given: �B � �D

Prove: ___

GB � ____

GD

Statements Reasons

1. How wide is the horse’s pasture?

PROBLEM 3 Applications


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2. Calculate AP if the perimeter of �AYP is 43 cm.

P

Y

A 13 cm

70°

70°

3. Lighting booms on a Ferris wheel consist of four steel beams that have cabling

with light bulbs attached. These beams along with 3 shorter beams form the

edges of three congruent isosceles triangles as shown. Maintenance crews

are installing new lighting along the four beams. Calculate the total length of

lighting needed.

4. Calculate m�T.

W M

117°

T


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5. What is the width of the river?

6. Given: ___

ST � ___

SR , ___

TA � ___

RA

Explain why �T � �R.

S

R

A

T


Lesson 6.5 | Isosceles Triangle Theorems 343

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OBJECTIVESIn this lesson you will:l Prove the Isosceles Triangle

Base Theorem.l Prove the Isosceles Triangle Vertex

Angle Theorem.l Prove the Isosceles Triangle

Perpendicular Bisector Theorem.l Prove the Isosceles Triangle Altitude to

Congruent Sides Theorem.l Prove the Isosceles Triangle Angle

Bisector to Congruent Sides Theorem.

KEY TERMSl vertex anglel Isosceles Triangle Base Theoreml Isosceles Triangle Vertex Angle Theoreml Isosceles Triangle Perpendicular

Bisector Theoreml Isosceles Triangle Altitude to Congruent

Sides Theoreml Isosceles Triangle Angle Bisector to

Congruent Sides Theorem

6.5 Isosceles Triangle TheoremsIsosceles Triangle Base Theorem, Vertex Angle Theorem, Perpendicular Bisector Theorem, Altitude to Congruent Sides Theorem, and Angle Bisector to Congruent Sides Theorem

PROBLEM 1 Isosceles Triangle TheoremsYou will prove theorems related to isosceles triangles. These proofs involve altitudes,

perpendicular bisectors, angle bisectors, and vertex angles. The vertex angle is the

angle formed by the two congruent legs in an isosceles triangle.

1. Given: Isosceles �ABC with ___

CA � ___

CB .

Construct altitude ___

CD from the vertex angle to the base.

A B

C

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a. Create a flow chart proof to prove ___

AD � ___

BD .

b. Create a two-column proof.

Given: Isosceles �ABC with ___

CA � ___

CB

Prove: ___

AD � ___

BD

Statements Reasons

Congratulations! You have just proven a theorem!

The Isosceles Triangle Base Theorem states: “The altitude to the base of an

isosceles triangle bisects the base.” You can now use this theorem as a valid reason

in proofs.


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2. In isosceles �ABC, with altitude ___

CD , explain how to prove �ACD � �BCD.

Congratulations! You have just explained how to prove another theorem!

The Isosceles Triangle Vertex Angle Theorem states: “The altitude to the base of

an isosceles triangle bisects the vertex angle.” You can now use this theorem as a

valid reason in proofs.

3. In isosceles �ABC, explain how to prove altitude ___

CD is the � bisector of ___

AB .

Congratulations! You have just explained how to prove another theorem!

The Isosceles Triangle Perpendicular Bisector Theorem states: “The altitude from

the vertex angle of an isosceles triangle is the perpendicular bisector of the base.”

You can now use this theorem as a valid reason in proofs.

The Isosceles Triangle Altitude to Congruent Sides Theorem states: “In an

isosceles triangle, the altitudes to the congruent sides are congruent.”

1. Draw and label a diagram for proving this theorem.

2. State the “Given” and “Prove” statements.

Given:

Prove:

PROBLEM 2 More Isosceles Triangle Theorems


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3. Write a paragraph proof of the Isosceles Triangle Altitude to Congruent

Sides Theorem.

4. Create a two-column proof of the Isosceles Triangle Altitude to Congruent

Sides Theorem.

Statements Reasons

The Isosceles Triangle Angle Bisector to Congruent Sides Theorem states:

“In an isosceles triangle, the angle bisectors to the congruent sides are congruent.”

5. Draw and label a diagram to prove this theorem.

6. State the “Given” and “Prove” statements.

Given:

Prove:


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7. Write a paragraph proof of the Isosceles Triangle Angle Bisector to Congruent

Sides Theorem.

8. Create a two-column proof of the Isosceles Triangle Angle Bisector to

Congruent Sides Theorem.

Statements Reasons


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1. Solve for the width of the dog house.

___

CD � ___

AB

___

AC � ___

BC

CD � 12�

AC � 20�

PROBLEM 4 Summary

PROBLEM 3 Dog House

Use the theorems you have just proven to answer each question about

isosceles triangles.

1. What can you conclude about an altitude drawn from the vertex angle to

the base?

2. What can you conclude about the altitudes to the congruent sides?

3. What can you conclude about the angle bisectors to the congruent sides?


Lesson 6.6 | Direct Proof vs. Indirect Proof 349

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6OBJECTIVESIn this lesson you will:l Write the inverse of a conditional

statement.l Differentiate between direct and

indirect proof.l Use indirect proof.

KEY TERMSl inversel contrapositivel direct proofl indirect proof or proof by contradictionl Hinge Theoreml Hinge Converse Theorem

6.6 Direct Proof vs. Indirect ProofInverse, Contrapositive, Direct Proof, and Indirect Proof

PROBLEM 1 The Inverse and ContrapositiveEvery conditional statement written in the form “If p, then q” has three additional

conditional statements associated with it: converse, contrapositive, and inverse.

Recall from previous lessons, to state the converse, reverse the hypothesis, p,

and the conclusion, q. To state the inverse, negate both parts. To state the

contrapositive, negate each part and reverse them.

Conditional Statement If p, then q.

Converse If q, then p.

Inverse If not p, then not q.

Contrapositive If not q, then not p.

For each conditional statement written in propositional form, identify the hypothesis p

and the conclusion q. Identify the negation of the hypothesis and conclusion, and

then write the inverse and contrapositive of the conditional statement.

1. If a quadrilateral is a square, then the quadrilateral is a rectangle.

a. Hypothesis p:

b. Conclusion q:

c. Is the conditional statement true? Explain.

d. Not p:

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e. Not q:

f. Inverse:

g. Is the inverse true? Explain.

h. Contrapositive:

i. Is the contrapositive true? Explain.

2. If an integer is even, then the integer is divisible by two.

a. Hypothesis p:

b. Conclusion q:


d. Not p:

e. Not q:

f. Inverse:


h. Contrapositive:



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3. If a polygon has six sides, then the polygon is a pentagon.

a. Hypothesis p:

b. Conclusion q:


d. Not p:

e. Not q:

f. Inverse:


h. Contrapositive:


4. If two lines intersect, then the lines are perpendicular.

a. Hypothesis p:

b. Conclusion q:


d. Not p:


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e. Not q:

f. Inverse:


h. Contrapositive:


5. What do you notice about the truth value of a conditional statement and the

truth value of its inverse?

6. What do you notice about the truth value of a conditional statement and the

truth value of its contrapositive?


6

All of the proofs up to this point were direct proofs. A direct proof begins with the

given information and works to the desired conclusion directly through the use of

givens, definitions, properties, postulates, and theorems.

An indirect proof is different and may be shorter than a direct proof. An indirect proof, or proof by contradiction, uses the contrapositive. If you prove the contrapositive

true, then the statement is true. Begin by assuming the conclusion is false and use

this assumption to show one of the given statements is false, thereby creating a

contradiction.

In an indirect proof:

• State the assumption; use the negation of the conclusion or prove statement.

• Write the givens.

• Write the negation of the conclusion.

• Use the assumption, in conjunction with definitions, properties, postulates, and

theorems, to prove a given statement is false, thus creating a contradiction.

Hence, your assumption leads to a contradiction; therefore, the assumption must be

false. This proves the contrapositive.

Let’s look at an example of an indirect proof.

Given: In �CHT, ___

CH � ___

CT

___

CA does not bisect ___

HT

Prove: �CHA � �CTA

H A

C

T

Statements Reasons

1. �CHA � �CTA 1. Assumption

2. ___

CA does not bisect ___

HT 2. Given

3. ___

HA � ___

TA 3. CPCTC

4. ___

CA bisects ___

HT 4. Defi nition of bisect

5. �CHA � �CTA is false 5. This is a contradiction.

Step 4 contradicts step 2;

the assumption is false

6. �CHA � �CTA is true 6. Proof by contradiction

In step 5, the “assumption” is stated as “false.” The reason for making this

statement is “contradiction.”

PROBLEM 2 Proof by Contradiction©

201

0 C

arne

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, Inc

.

Take NoteNotice, you are

trying to prove

�CHA � �CTA.

You assume the

negation of this

statement,

�CHA � �CTA.

This becomes the

first statement in

your proof and the

reason for making

this statement is

“assumption.”


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Now try one yourself!

1. Given: ___

BR bisects �ABN A

B

R

N

�BRA � �BRN

Prove: ___

AB � ___

NB

Statements Reasons

2. When writing an indirect proof, it is often easier to write it as a paragraph proof.

Write the proof in Question 1 as a paragraph proof.

3. Use a paragraph proof model to write an indirect proof proving a triangle

cannot have more than one right angle.


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The Hinge Theorem states: “If two sides of one triangle are congruent to two sides

of another triangle and the included angle of the first pair is larger than the included

angle of the second pair, then the third side of the first triangle is longer than the

third side of the second triangle.”

In the two triangles shown, notice that RS � DE, ST � EF, and �S � �E. The Hinge

Theorem guarantees that RT � DE.

R

S T100°

D

E F80°

1. Use an indirect proof to prove the Hinge Theorem.

Begin by restating the Hinge Theorem using �ABC and �DEF.

If sides AB � DE and AC � DF, and the included angle at A is larger than the

included angle at D, then BC � EF.

A

B

C

D

EF

Given: AB � DE

AC � DF

m�A � m�D

Prove: BC � EF

PROBLEM 3 Hinge Theorem and Its Converse


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This theorem must be proven for two cases:

Case 1: BC � EF

Case 2: BC � EF

a. Write the indirect proof for Case 1.

Statements Reasons

b. Write the indirect proof for Case 2.

Statements Reasons


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The Hinge Converse Theorem states: “If two sides of one triangle are congruent to

two sides of another triangle and the third side of the first triangle is longer than the

third side of the second triangle, then the included angle of the first pair of sides is

larger than the included angle of the second pair of sides.”

In the two triangles shown, notice that RT � DF, RS � DE, and ST � EF. The Hinge

Converse Theorem guarantees that m�R � m�D.

S

R

T

10

E

D

8

F

2. Create an indirect proof to prove the Hinge Converse Theorem.

A

C

B

D

E F

Given: AB � DE

AC � DF

BC � EF

Prove: m�A � m�D

This theorem must be proven for two cases:

Case 1: m�A � m�D

Case 2: m�A � m�D

a. Create an indirect proof for Case 1.

Statements Reasons


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b. Create an indirect proof for Case 2.

Statements Reasons

3. Use the Hinge Theorem and its converse to answer each.

a. Matthew and Jeremy’s families are going

camping for the weekend. Before heading out

of town, they decide to meet at Al’s Diner for

breakfast. During breakfast, the boys try to

decide which family will be further away from

the diner “as the crow flies.” “As the crow flies”

is an expression based on the fact that crows,

generally fly straight to the nearest food supply.

Matthew’s family is driving 35 miles due north

and taking an exit to travel an additional

15 miles northeast. Jeremy’s family is driving

35 miles due south and taking an exit to travel

an additional 15 miles southwest. Use the

diagram shown to determine which family is

further from the diner. Explain your reasoning.


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b. Which of the following is a possible length for AH: 20 cm, 21 cm, or 24 cm?

Explain your choice.

55°

21 cm

W

E P

61°

A

HR

c. Which of the following is a possible angle measure for �ARH: 54º, 55º or

56º? Explain your choice.

55°

34 mm

W

PE

A

HR

36 mm



6KEY TERMSl corresponding parts of

congruent triangles are congruent (CPCTC) (6.4)

l angle bisector (6.5)

l perpendicular bisector (6.5)l vertex angle (6.5)l inverse (6.6)l contrapositive (6.6)

l direct proof (6.6)l indirect proof or proof by

contradiction (6.6)

THEOREMSl Side-Side-Side (SSS)

Congruence Theorem (6.2)l Side-Angle-Side (SAS)

Congruence Theorem (6.2)l Angle-Side-Angle (ASA)

Congruence Theorem (6.2)l Angle-Angle-Side (AAS)

Congruence Theorem (6.2)l Hypotenuse-Leg (HL)

Congruence Theorem (6.3)l Leg-Leg (LL) Congruence

Theorem (6.3)

l Hypotenuse-Angle (HA) Congruence Theorem (6.3)

l Leg-Angle (LA) Congruence Theorem (6.3)

l Isosceles Triangle Base Angle Theorem (6.4)

l Isosceles Triangle Base Angle Converse Theorem (6.4)

l Isosceles Triangle Base Theorem (6.5)

l Isosceles Triangle Vertex Angle Theorem (6.5)

l Isosceles Triangle Perpendicular Bisector Theorem (6.5)

l Isosceles Triangle Altitude to Congruent Sides Theorem (6.5)

l Isosceles Triangle Angle Bisector to Congruent Sides Theorem (6.5)

l Hinge Theorem (6.6)l Hinge Converse

Theorem (6.6)

CONSTRUCTIONSl triangle given two line

segments (6.1)l triangle given three line

segments (6.1)

l triangle given two line segments and the included angle (6.1)

l triangle given three angles (6.1)

l triangle given two angles and one line segment (6.1)

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Chapter 6 Checklist

Chapter 6 | Checklist 361

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Constructing Triangles

By constructing triangles, you discovered that all combinations of givens that

determine a unique triangle are:

1. Given three sides

2. Given two sides and the included angle

3. Given two angles and a specific side

You discovered that all combinations of givens that determine multiple triangles are:

1. Given two sides

2. Given two or three angles

3. Given two sides and an angle not included

4. Given two angles and a side not specified

Using the Side-Side-Side (SSS) Congruence Theorem

The Side-Side-Side (SSS) Congruence Theorem states: “If three sides of one triangle

are congruent to the corresponding sides of another triangle, then the triangles are

congruent.”

Example:

R

S

T

5 m

6 m

7 m

F

G

H5 m

6 m

7 m

___

FG � ___

RS , ___

GH � ___

ST , and ___

FG � ___

RT , so �FGH � �RST.

6.1

6.2

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Using the Side-Angle-Side (SAS) Congruence Theorem

The Side-Angle-Side (SAS) Congruence Theorem states: “If two sides and the

included angle of one triangle are congruent to the corresponding two sides and the

included angle of another triangle, then the triangles are congruent.”

Example:

X

10 in.

Y

Z

17 in.44°

L

M

N

10 in.

17 in.

44°

___

XY � ___

LM , �Y � �M, and ___

YZ � ____

MN , so �XYZ � �LMN.

Using the Angle-Side-Angle (ASA) Congruence Theorem

The Angle-Side-Angle (ASA) Congruence Theorem states: “If two angles and the

included side of one triangle are congruent to the corresponding two angles and the

included side of another triangle, the triangles are congruent.”

Example:

J K

L

15 cm

87° 62°

D

E F62°

87°15 cm

�D � �J, ___

DE � ___

JK , and �E � �K, so �DEF � �JKL.

Using the Angle-Angle-Side (AAS) Congruence Theorem

The Angle-Angle-Side (AAS) Congruence Theorem states: “If two angles and the

non-included side of one triangle are congruent to the corresponding two angles and

the non-included side of another triangle, the triangles are congruent.”

Example:

P

Q

R

2 ft

35°

115°

V

W

X

115° 35°2 ft

�P � �V, �Q � �W, and ___

QR � ____

WK , so �PQR � �VWX.

6.2

6.2

6.2


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Using the Hypotenuse-Leg (HL) Congruence Theorem

The Hypotenuse-Leg (HL) Congruence Theorem states: “If the hypotenuse and leg of

one right triangle are congruent to the hypotenuse and leg of another right triangle,


Example:

6 in.

A

B

C3 in.

F

D

E6 in.

3 in.

___

BC � ___

EF , ___

AC � ___

DF , and angles A and D are right angles, so �ABC � �DEF.

Using the Leg-Leg (LL) Congruence Theorem

The Leg-Leg (LL) Congruence Theorem states: “If two legs of one right triangle are

congruent to two legs of another right triangle, then the triangles are congruent.”

Example:

12 ft

11 ftX

Y

Z

12 ft

11 ft

RS

T

___

XY � ___

RS , ___

XZ � ___

RT , and angles X and R are right angles, so �XYZ � �RST.

6.3

6.3

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Using the Hypotenuse-Angle (HA) Congruence Theorem

The Hypotenuse-Angle (HA) Congruence Theorem states: “If the hypotenuse and an

acute angle of one right triangle are congruent to the hypotenuse and an acute angle

of another right triangle, then the triangles are congruent.”

Example:

D

E

FJ

K

L

10 m

32°

32°10 m

___

KL � ___

EF , �L � �F, and angles J and D are right angles, so �JKL � �DEF.

Using the Leg-Angle (LA) Congruence Theorem

The Leg-Angle (LA) Congruence Theorem states: “If a leg and an acute angle of one

right triangle are congruent to the leg and an acute angle of another right triangle,


Example:

L

M

N

9 mm

51°

9 mm

51°

G

JH

____

GN � ___

LN , �H � �M, and angles G and L are right angles,

so �GHJ � �LMN.

6.3

6.3


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Using CPCTC to Solve a Problem

If two triangles are congruent, then each part of one triangle is congruent to the

corresponding part of the other triangle. In other words, “corresponding parts of

congruent triangles are congruent,” which is abbreviated CPCTC. To use CPCTC,

first prove that two triangles are congruent.

Example:

You want to determine the distance between two docks along a river. The docks are

represented as points A and B in the diagram below. You place a marker at point X,

because you know that the distance between points X and B is 26 feet. Then you

walk horizontally from point X and place a marker at point Y, which is 26 feet from

point X. You measure the distance between points X and A to be 18 feet, and so you

walk along the river bank 18 feet and place a marker at point Z. Finally, you measure

the distance between Y and Z to be 35 feet.

From the diagram, segments XY and XB are congruent and segments XA and XZ

are congruent. Also, angles YXZ and BXA are congruent by the Vertical Angles

Congruence Theorem. So, by the Side-Angle-Side (SAS) Congruence Postulate,

�YXZ � �BXA. Because corresponding parts of congruent triangles are congruent

(CPCTC), segment YZ must be congruent to segment BA. The length of segment YZ is

35 feet. So, the length of segment BA, or the distance between the docks, is 35 feet.

6.4

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Using the Isosceles Triangle Base Angle Theorem

The Isosceles Triangle Base Angle Theorem states: “If two sides of a triangle are

congruent, then the angles opposite these sides are congruent.”

Example:

F40°

15 yd

G

H

15 yd

___

FH � ____

GH , so �F � �G, and the measure of angle G is 40°.

Using the Isosceles Triangle Base Angle Converse Theorem

The Isosceles Triangle Base Angle Converse Theorem states: “If two

angles of a triangle are congruent, then the sides opposite these angles

are congruent.”

Example:

J

75°

21 m

L

K

75°

�J � �K, ___

JL � ___

KL , and the length of side KL is 21 meters.

Using the Isosceles Triangle Base Theorem


isosceles triangle bisects the base.”

Example:

A

100 ft

C

B

100 ft

D75 ft x

CD � AD, so x � 75 feet.

6.4

6.4

6.5


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Using the Isosceles Triangle Vertex Angle Theorem


isosceles triangle bisects the vertex angle.”

Example:

F

5 in.

J

H

5 in.

G48°

x

m�FGJ � m�HGJ, so x � 48°.

Using the Isosceles Triangle Perpendicular Bisector Theorem

The Isosceles Triangle Perpendicular Bisector Theorem states: “The altitude from the

vertex angle of an isosceles triangle is the perpendicular bisector of the base.”

Example:

W Z Y

X

____

WY � ___

XZ and WZ � YZ

6.5

6.5

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Using the Isosceles Triangle Altitude to Congruent Sides Theorem

The Isosceles Triangle Perpendicular Bisector Theorem states: “In an isosceles

triangle, the altitudes to the congruent sides are congruent.”

Example:

J 11 m

L

K11 m

M

N

___

KN � ___

JM

Using the Isosceles Triangle Bisector to Congruent Sides Theorem

The Isosceles Triangle Perpendicular Bisector Theorem states: “In an isosceles

triangle, the angle bisectors to the congruent sides are congruent.”

Example:

R

12 cm

T

S

12 cm

V W

____

RW � ___

TV

Stating the Inverse and Contrapositive of Conditional Statements

To state the inverse of a conditional statement, negate both the hypothesis and the

conclusion. To state the contrapositive of a conditional statement, negate both the

hypothesis and the conclusion and then reverse them.

Conditional Statement: If p, then q.

Inverse: If not p, then not q.

Contrapositive: If not q, then not p.

Example:

Conditional Statement: If a triangle is equilateral, then it is isosceles.

Inverse: If a triangle is not equilateral, then it is not isosceles.

Contrapositive: If a triangle is not isosceles, then it is not equilateral.

6.5

6.5

6.6


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Writing an Indirect Proof

In an indirect proof, or proof by contradiction, first write the givens. Then, write the

negation of the conclusion. Then, use that assumption to prove a given statement is

false, thus creating a contradiction. Hence, the assumption leads to a contradiction,

therefore showing that the assumption is false. This proves the contrapositive.

Example:

Given: Triangle DEF

Prove: A triangle cannot have more than one obtuse angle.

Given �DEF, assume that �DEF has two obtuse angles. So, assume m�D � 91°

and m�E � 91°. By the Triangle Sum Theorem, m�D � m�E � m�F � 180°.

By substitution, 91° � 91° � m�F � 180°, and by subtraction, m�F � 2°.

But it is not possible for a triangle to have a negative angle, so this is a contradiction.

This proves that a triangle cannot have more than one obtuse angle.

Using the Hinge Theorem

The Hinge Theorem states: “If two sides of one triangle are congruent to two sides

of another triangle and the included angle of the first pair is larger than the included

angle of the second pair, then the third side of the first triangle is longer than the

third side of the second triangle.”

Example:

x

P80° 75°

Q

R F

G

H

8 mm

QR � GH, so x � 8 millimeters.

6.6

6.6

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Using the Hinge Converse Theorem

The Hinge Converse Theorem states: “If two sides of one triangle are congruent to

two sides of another triangle and the third side of the first triangle is longer than the

third side of the second triangle, then the included angle of the first pair of sides is

larger than the included angle of the second pair of sides.”

Example:

xX 62°

Y

ZR

S

T

5 ft 4 ft

m�T � m�Z, so x � 62°.

6.6

Documents

Congruence - Oak Park Geometry Classes - Homeophsgc.weebly.com/uploads/1/3/2/9/13299817/geometry... · · 2013-04-26SSS, SAS, ASA, and AAS | p. 319 6.3 p. 343Right Triangle Congruence