Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Warm Up

1. If ∆ABC ∆DEF, then A ? and BC ? .

2. What is the distance between (3, 4) and (–1, 5)?

3. If 1 2, why is a||b?

4. List methods used to prove two triangles congruent.

D EF

17

Converse of Alternate Interior Angles Theorem

SSS, SAS, and ASA Postulates, AAS and HL Theorems

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Use CPCTC to prove parts of triangles are congruent.

Learning Target

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

CPCTC – Corresponding Parts of Congruent Triangles are Congruent

Vocabulary

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

CPCTC is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are Congruent.” It can be used as a justification in a proof after you have proven two triangles congruent.

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

SSS, SAS, and ASA Postulates, and AAS and HL Theorems use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent.

Remember!

You can only use CPCTC AFTER you have proven two triangles congruent.

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Example 1: Engineering Application

A and B are on the edges of a ravine. What is AB? One angle pair is congruent, because they are vertical angles. Two pairs of sides are congruent, because their lengths are equal.

Therefore the two triangles are congruent by SAS Postulate. By CPCTC, the third side pair is congruent, so AB = 18 mi.

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Check It Out! Example 1

A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles.

Two pairs of sides are congruent, because their lengths are equal. Therefore the two triangles are congruent by SAS. By CPCTC, the third side pair is congruent, so JK = 41 ft.

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Example 2: Proving Corresponding Parts Congruent

Prove: XYW ZYW

Given: YW bisects XZ, XY ZY.

Z

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Example 2 Continued

WY

ZW

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Statements Reasons

1. bisects , 1. Given

2. 2. Def. Segment Bisector

3. 3. Reflexive Property of

YW XZ XY ZY

XW ZW

YW YW

4. 4. SSS Postulate

5. 5. CPCTC

XYW ZYW

XYW ZYW

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Check It Out! Example 2

Prove: PQ PS

Given: PR bisects QPS and QRS.

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Check It Out! Example 2 Continued

PR bisects QPS

and QRS

QRP SRP

QPR SPR

Given Def. of bisector

RP PR

Reflex. Prop. of

∆PQR ∆PSR

PQ PS

ASA

CPCTC

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

1. bisects and 1. Given

2. , 2. Def. Angle Bisector

3. 3. Reflexive Property of

4.

PR QPS QRS

QPR SPR QRP SRP

RP RP

QPR SPR

4. ASA Postulate

5. 5. CPCTCPQ PS

Statements Reasons

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Work backward when planning a proof. To show that ED || GF, look for a pair of angles that are congruent.

Then look for triangles that contain these angles.

Helpful Hint

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Example 3: Using CPCTC in a Proof

Prove: MN || OP

Given: NO || MP, N P

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

5. CPCTC5. NMO POM

6. Conv. Of Alt. Int. s Thm.

4. AAS Theorem4. ∆MNO ∆OPM

3. Reflex. Prop. of

2. Alternate Interior Angles Theorem.2. NOM PMO

1. Given

ReasonsStatements

3. MO MO

6. MN || OP

1. N P; NO || MP

Example 3 Continued

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Check It Out! Example 3

Prove: KL || MN

Given: J is the midpoint of KM and NL.

Holt McDougal Geometry

4-7 Triangle Congruence: CPCTC

Homework: pg 270-271, #3, 4, 7-18