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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 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