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Warm Up 1. If ∆ ABC  ∆ DEF , then  A  ? and BC  ? .

EF.  17. 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. Converse of Alternate Interior Angles Theorem.

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Warm Up 1. If ∆ ABC  ∆ DEF , then  A  ? and BC  ? .

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  1. EF 17 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 Converse of Alternate Interior Angles Theorem SSS, SAS, and ASA Postulates, AAS and HL Theorems

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

  3. Vocabulary CPCTC – Corresponding Parts of Congruent Triangles are Congruent

  4. CPCTCis 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.

  5. Remember! 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. You can only use CPCTC AFTER you have proven two triangles congruent.

  6. Therefore the two triangles are congruent by SAS Postulate. By CPCTC, the third side pair is congruent, so AB = 18 mi. 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.

  7. 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.

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

  9. ZW WY Example 2 Continued

  10. Statements Reasons

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

  12. QRP SRP QPR  SPR PR bisects QPS and QRS RP PR Reflex. Prop. of  Def. of  bisector Given ∆PQR  ∆PSR ASA PQPS CPCTC Check It Out! Example 2 Continued

  13. Statements Reasons

  14. 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.

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

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

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

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

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