1. 4-6 Warm Up Lesson Presentation Lesson Quiz Triangle Congruence: CPCTC
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz

2. Do Now 1. If ∆ABC  ∆DEF, then A  ? and BC  ? .
2. If 1  2, why is a||b?

3. Objective
Use CPCTC to prove parts of triangles are congruent.

4. Vocabulary
CPCTC

5. 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.

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

7. Example 1: Engineering Application
A and B are on the edges of a ravine. What is AB?

8. Example 2
A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK?

9. Example 3: Proving Corresponding Parts Congruent
Given: YW bisects XZ, XY  YZ. Z
Prove: XYW  ZYW
Statements
Reasons
1. 𝑌𝑊 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝑋𝑍
1. Given
2. 𝑋𝑌  𝑌𝑍
2. Given
3. 𝑋𝑊  𝑍𝑊
3. Def. segment bisector
4. 𝑌𝑊  𝑌𝑊
4. Reflexive POC
5. 𝑋𝑌𝑊 ZYW
5. SSS
6. 𝑋𝑌𝑊ZYW
6. CPCTC

10. 1. 𝑃𝑅 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝑄𝑃𝑆 𝑎𝑛𝑑  QRS
Example 4
Prove: PQ  PS
Given: PR bisects QPS and QRS.
Statements
Reasons
1. 𝑃𝑅 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝑄𝑃𝑆 𝑎𝑛𝑑  QRS
1. Given
2.𝑄𝑃𝑅SPR
2. Def.  bisector
3.𝑄𝑅𝑃SRP
3. Def.  bisector
4. 𝑃𝑅  𝑃𝑅
4. Reflexive POC
5. 𝑄𝑃𝑅 SPR
5. ASA
6. 𝑃𝑄  𝑃𝑆
6. CPCTC

11. Then look for triangles that contain these angles.
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

12. Example 5: Using CPCTC in a Proof
Prove: MN || OP
Given: NO || MP, N  P 1 3 4 2
Statements
Reasons
1. 𝑁𝑂 𝑀𝑃
1. Given
2.𝑁O
2. Given
3.12
3. Alt. int.  th.
4. 𝑀𝑂  𝑀𝑂
4. Reflexive POC
5. 𝑁𝑂𝑀 PMO
5. AAS
6. CPCTC
6.34
7. 𝑀𝑁 𝑂𝑃
7. Converse of alt. int.  th.

13. Given: J is the midpoint of KM and NL.
Example 6
Prove: KL || MN
Given: J is the midpoint of KM and NL. 3 1
Statements
Reasons 2 4
1. 𝐽 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐾𝑀 𝑎𝑛𝑑 𝑁𝐿
1. Given
2. 𝐾𝐽  𝑀𝐽
2. Def. midpoint
3. 𝑁𝐽  𝐿𝐽
3. Def. midpoint
4.12
4. Vertical  th.
5. 𝐾𝐽𝐿 MJN
5. SAS
6. CPCTC
6.34
7. 𝐾𝐿 𝑀𝑁
7. Converse of alt. int.  th.

14. You Try It!
Given: X is the midpoint of AC . 1  2 Prove: X is the midpoint of BD. 4 3
Statements
Reasons
1. 𝑋 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐶
1. Given
2.12
2. Given
3. 𝐴𝑋  𝐶𝑋
3. Def. midpoint
4.34
4. Vertical  th.
5. 𝐴𝑋𝐷 CXB
5. ASA
6. CPCTC
6. 𝐷𝑋  𝐵𝑋
7. 𝑋 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐵𝐷
7. Def. midpoint