Holt Geometry

4-6 Triangle Congruence: CPCTC

Use CPCTC to prove parts of triangles are congruent.

Objective

Holt Geometry

4-6 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 Geometry

4-6 Triangle Congruence: CPCTC

You much prove SSS, SAS, ASA, AAS, or HL first, and THEN use CPCTC to show that corresponding parts of triangles are congruent.

Remember!

Holt Geometry

4-6 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. By CPCTC, the third side pair is congruent, so AB = 18 mi.

Holt Geometry

4-6 Triangle Congruence: CPCTC

Example 2: Proving Corresponding Parts Congruent

Prove: XYW ZYW

Given: YW bisects XZ, XY YZ.

Z

Holt Geometry

4-6 Triangle Congruence: CPCTC

Example 2 Continued

WY

ZW

Holt Geometry

4-6 Triangle Congruence: CPCTC

Check It Out! Example 2

Prove: PQ PS

Given: PR bisects QPS and QRS.

Holt Geometry

4-6 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 Geometry

4-6 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 Geometry

4-6 Triangle Congruence: CPCTC

Example 3: Using CPCTC in a Proof

Prove: MN || OP

Given: NO || MP, N P

Holt Geometry

4-6 Triangle Congruence: CPCTC

5. CPCTC5. NMO POM

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

4. AAS4. ∆MNO ∆OPM

3. Reflex. Prop. of

2. Alt. Int. s Thm.2. NOM PMO

1. Given

ReasonsStatements

3. MO MO

6. MN || OP

1. N P; NO || MP

Example 3 Continued

Holt Geometry

4-6 Triangle Congruence: CPCTC

Lesson Quiz: Part II

2. Given: X is the midpoint of AC . 1 2

Prove: X is the midpoint of BD.

Holt Geometry

4-6 Triangle Congruence: CPCTC

Lesson Quiz: Part II Continued

6. CPCTC

7. Def. of 7. DX = BX

5. ASA Steps 1, 4, 55. ∆AXD ∆CXB

8. Def. of mdpt.8. X is mdpt. of BD.

4. Vert. s Thm.4. AXD CXB

3. Def of 3. AX CX

2. Def. of mdpt.2. AX = CX

1. Given1. X is mdpt. of AC. 1 2

ReasonsStatements

6. DX BX