advertisement

4-6 TriangleCongruence: Congruence: CPCTC CPCTC 4-6 Triangle Warm Up Lesson Presentation Lesson Quiz Holt Geometry 4-6 Triangle Congruence: CPCTC Do Now 1. If ∆ABC ∆DEF, then A 2. If 1 2, why is a||b? Holt Geometry ? and BC ? . 4-6 Triangle Congruence: CPCTC Objective Use CPCTC to prove parts of triangles are congruent. Holt Geometry 4-6 Triangle Congruence: CPCTC Vocabulary CPCTC 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 Remember! SSS, SAS, ASA, AAS, and HL use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. 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 Given: YW bisects XZ, XY YZ. Prove: XYW ZYW Statements Reasons 1. 𝑌𝑊𝑏𝑖𝑠𝑒𝑐𝑡𝑠 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. 𝑋𝑌𝑊ZYW 6. CPCTC Holt Geometry Z 4-6 Triangle Congruence: CPCTC Example 4 Given: PR bisects QPS and QRS. Prove: PQ PS Statements Reasons 1. 𝑃𝑅 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝑄𝑃𝑆 𝑎𝑛𝑑 QRS 1. Given 2. 𝑄𝑃𝑅SPR 2. Def. bisector 3. 𝑄𝑅𝑃SRP 3. Def. bisector 4. 𝑃𝑅 𝑃𝑅 4. Reflexive POC 5. 𝑄𝑃𝑅 SPR 5. ASA 6. 𝑃𝑄 𝑃𝑆 6. CPCTC Holt Geometry 4-6 Triangle Congruence: CPCTC Helpful Hint 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. Holt Geometry 4-6 Triangle Congruence: CPCTC Example 5: Using CPCTC in a Proof Given: NO || MP, N P Prove: MN || OP Statements 1 4 2 Reasons 1. 𝑁𝑂 𝑀𝑃 1. Given 2. 𝑁O 2. Given 3. 12 3. Alt. int. th. 4. 𝑀𝑂 𝑀𝑂 4. Reflexive POC 5. 𝑁𝑂𝑀 PMO 5. AAS 6. 34 6. CPCTC 7. 𝑀𝑁 𝑂𝑃 7. Converse of alt. int. th. Holt Geometry 3 4-6 Triangle Congruence: CPCTC Example 6 Given: J is the midpoint of KM and NL. Prove: KL || MN 3 1 Statements Reasons 2 4 1. 𝐽 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐾𝑀 𝑎𝑛𝑑𝑁𝐿 1. Given 2. 𝐾𝐽 𝑀𝐽 2. Def. midpoint 3. 𝑁𝐽 𝐿𝐽 3. Def. midpoint 4. 12 4. Vertical th. 5. 𝐾𝐽𝐿 MJN 5. SAS 6. 34 6. CPCTC 7. 𝐾𝐿 𝑀𝑁 7. Converse of alt. int. th. Holt Geometry 4-6 Triangle Congruence: CPCTC You Try It! Given: X is the midpoint of AC . 1 2 Prove: X is the midpoint of BD. Statements 1. 𝑋 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐶 2. 12 Reasons 1. Given 2. Given 3. 𝐴𝑋 𝐶𝑋 3. Def. midpoint 4. 34 4. Vertical th. 5. 𝐴𝑋𝐷 CXB 5. ASA 6. 𝐷𝑋 𝐵𝑋 6. CPCTC 7. 𝑋 𝑖𝑠 𝑡ℎ𝑒 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐵𝐷 7. Def. midpoint Holt Geometry 4 3