Warm Up Lesson Presentation Lesson Quiz Triangle Congruence: CPCTC Holt Geometry Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry

Warm-up Identify the postulate or theorem that proves the triangles congruent. HL ASA SAS or SSS

4. Given: PN bisects MO, PN  MO Warm-up 4. Given: PN bisects MO, PN  MO Prove: ∆MNP  ∆ONP 1. Given 2. Def. of bisect 3. Reflex. Prop. of  4. Given 5. Def. of  6. Rt.   Thm. 7. SAS 1. PN bisects MO 2. MN  ON 3. PN  PN 4. PN  MO 5. PNM and PNO are rt. s 6. PNM  PNO 7. ∆MNP  ∆ONP Reasons Statements

Objective Use CPCTC to prove parts of triangles are congruent.

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.

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

Example 2: Proving Corresponding Parts Congruent Given: YW bisects XZ, XY  YZ. Prove: XYW  ZYW Z

Example 2 Continued WY ZW

Given: PR bisects QPS and QRS. Check It Out! Example 2 Prove: PQ  PS Given: PR bisects QPS and QRS.

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

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

Example 3: Using CPCTC in a Proof Prove: MN || OP Given: NO || MP, N  P

Example 3 Continued Statements Reasons 1. N  P; NO || MP 1. Given 2. NOM  PMO 2. Alt. Int. s Thm. 3. MO  MO 3. Reflex. Prop. of  4. ∆MNO  ∆OPM 4. AAS 5. NMO  POM 5. CPCTC 6. MN || OP 6. Conv. Of Alt. Int. s Thm.

Given: J is the midpoint of KM and NL. Check It Out! Example 3 Prove: KL || MN Given: J is the midpoint of KM and NL.

Check It Out! Example 3 Continued Statements Reasons 1. Given 1. J is the midpoint of KM and NL. 2. KJ  MJ, NJ  LJ 2. Def. of mdpt. 3. KJL  MJN 3. Vert. s Thm. 4. ∆KJL  ∆MJN 4. SAS 5. LKJ  NMJ 5. CPCTC 6. KL || MN 6. Conv. Of Alt. Int. s Thm.

Lesson Quiz: Part I 1. Given: Isosceles ∆PQR, base QR, PA  PB Prove: AR  BQ

Lesson Quiz: Part I Continued 4. Reflex. Prop. of  4. P  P 5. SAS 5. ∆QPB  ∆RPA 6. CPCTC 6. AR = BQ 3. Given 3. PA = PB 2. Def. of Isosc. ∆ 2. PQ = PR 1. Isosc. ∆PQR, base QR Statements 1. Given Reasons

2. Given: X is the midpoint of AC . 1  2 Lesson Quiz: Part II 2. Given: X is the midpoint of AC . 1  2 Prove: X is the midpoint of BD.

Lesson Quiz: Part II Continued 5. CPCTC 4. ASA 4. ∆AXD  ∆CXB 6. Def. of mdpt. 6. X is mdpt. of BD. 3. Vert. s Thm. 3. AXD  CXB 2. Def. of midpt. 2. AX  CX 1. Given 1. X is mdpt. of AC. 1  2 Reasons Statements 5. DX  BX