Warm Up 1. If ∆ABC  ∆DEF, then A  ? and BC  ? . 2. What is the distance between (3, 4) and (–1, 5)? 3. If 1  2, why is a||b? 4. List methods used to prove two triangles congruent. D EF 17 Converse of Alternate Interior Angles Theorem SSS, SAS, and ASA Postulates, AAS and HL Theorems

Learning Target Use CPCTC to prove parts of triangles are congruent.

Vocabulary CPCTC – Corresponding Parts of Congruent 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.

You can only use CPCTC AFTER you have proven two triangles congruent. SSS, SAS, and ASA Postulates, and AAS and HL Theorems use corresponding parts to prove triangles congruent. CPCTC uses congruent triangles to prove corresponding parts congruent. Remember! You can only use CPCTC AFTER you have proven two triangles congruent.

Example 1: Engineering Application A and B are on the edges of a ravine. What is AB? One angle pair is congruent, because they are vertical angles. Two pairs of sides are congruent, because their lengths are equal. Therefore the two triangles are congruent by SAS Postulate. By CPCTC, the third side pair is congruent, so AB = 18 mi.

Check It Out! Example 1 A landscape architect sets up the triangles shown in the figure to find the distance JK across a pond. What is JK? One angle pair is congruent, because they are vertical angles. Two pairs of sides are congruent, because their lengths are equal. Therefore the two triangles are congruent by SAS. By CPCTC, the third side pair is congruent, so JK = 41 ft.

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

Example 2 Continued WY ZW

Statements Reasons

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

Statements Reasons

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. Alternate Interior Angles Theorem. 3. MO  MO 3. Reflex. Prop. of  4. ∆MNO  ∆OPM 4. AAS Theorem 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.

Homework: pg 270-271, #3, 4, 7-18