Holt Geometry

4-6 Triangle Congruence: CPCTC4-6 Triangle Congruence: CPCTC

Holt Geometry

Warm Up

Lesson Presentation

Lesson Quiz

Holt Geometry

4-6 Triangle Congruence: CPCTC

Do Now

1. If ∆ABC ∆DEF, then A ? and BC ? .

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

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

Vocabulary

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

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

Holt Geometry

4-6 Triangle Congruence: CPCTC

Example 2

A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK?

Holt Geometry

4-6 Triangle Congruence: CPCTC

Example 3: Proving Corresponding Parts Congruent

Prove: XYW ZYW Given: YW bisects XZ, XY YZ.

Z

Statements Reasons1.𝑌𝑊𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝑋𝑍

2. 𝑋𝑌 𝑌𝑍

3. 𝑋𝑊 𝑍𝑊

6. 𝑋𝑌𝑊ZYW

1. Given2. Given3. Def. segment bisector4. Reflexive POC4.𝑌𝑊 𝑌𝑊 5. SSS6. CPCTC

Holt Geometry

4-6 Triangle Congruence: CPCTC

Prove: PQ PS Given: PR bisects QPS and QRS. Example 4

Statements Reasons QRS

2.𝑄𝑃𝑅SPR

3.𝑄𝑅𝑃SRP

6.𝑃𝑄𝑃𝑆

1. Given2. Def. bisector3. Def. bisector4. Reflexive POC4.𝑃𝑅𝑃𝑅 5. ASA6. 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 5: Using CPCTC in a Proof

Prove: MN || OP

Given: NO || MP, N P

Statements Reasons1.𝑁𝑂‖𝑀𝑃2.𝑁O3.12

6.3 4

1. Given2. Given3. Alt. int. th.4. Reflexive POC4.𝑀𝑂𝑀𝑂 5. AAS6. CPCTC

1

2

3

4

7.𝑀𝑁‖𝑂𝑃 7. Converse of alt. int. th.

Holt Geometry

4-6 Triangle Congruence: CPCTC

Example 6

Prove: KL || MN

Given: J is the midpoint of KM and NL.

Statements Reasons1. 𝐽 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐾𝑀 𝑎𝑛𝑑𝑁𝐿2.𝐾𝐽𝑀 𝐽3.𝑁 𝐽 𝐿 𝐽

6.3 4

1. Given2. Def. midpoint3. Def. midpoint4. Vertical th.4.1 2 5. SAS6. CPCTC7.𝐾𝐿‖𝑀𝑁 7. Converse of alt. int.

th.

2

1

4

3

Holt Geometry

4-6 Triangle Congruence: CPCTC

You Try It!Given: X is the midpoint of AC . 1 2Prove: X is the midpoint of BD.

Statements Reasons1. 𝑋𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐶

3. 𝐴𝑋 𝐶𝑋

6.𝐷 𝑋𝐵 𝑋

1. Given2. Given3. Def. midpoint4. Vertical th.4.3 4 5. ASA6. CPCTC7. 𝑋 𝑖𝑠 h𝑡 𝑒𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐵𝐷 7. Def. midpoint

4 3