1. Using Congruent Triangles Section 5.5

2. Objective Show corresponding parts of congruent triangles are congruent.

3. Key Vocabulary - Review Corresponding parts

4. Review: Congruence Shortcuts

5. Congruent Triangles (CPCTC) congruent triangles cp ct c Two triangles are congruent triangles if and only if the corresponding parts of those congruent triangles are congruent. Corresponding sides are congruent Corresponding angles are congruent

6. Example: Name the Congruence Shortcut or CBD SAS ASA SSS SSA CBD

7. Your Turn: Name the Congruence Shortcut or CBD ASA SAS AAA SSA CBD CBD

8. SAS SAS SAS Reflexive Property Vertical Angles Reflexive Property SSA CBD

9. Your Turn: Name the Congruence Shortcut or CBD SSS ASA SSA CBD AAA CBD

10. ASA SSA CBD SAS

11. Example Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS:  B   D For AAS: A  F A  F AC  FE

12. Your Turn: Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS: N  V N  V MK  UT M  U M  U

13. Using Congruent Triangles: CPCTC If you know that two triangles are congruent, then you can use CPCTC to prove the corresponding parts in whose triangles are congruent. *You must prove that the triangles are congruent before you can use CPCTC*

14. Example 1 Use Corresponding Parts In the diagram, AB and CD bisect each other at M. Prove that  A   B. Because  A and  B are corresponding angles in ∆ADM and ∆BCM, show that ∆ADM  ∆BCM to prove that  A   B. 2. SOLUTION First sketch the diagram and label any congruent segments and congruent angles. 1.

15. Example 1 Use Corresponding Parts Statements Reasons 1. AB and CD bisect each other at M. Given1. Definition of segment bisector2. MA  MB  AMD   BMC 3.Vertical Angles Theorem3. ∆ADM  ∆BCM 5.SAS Congruence Postulate5.  A   B 6.Corresponding parts of congruent triangles are congruent. 6. Definition of segment bisector4. MD  MC

16. The Proof Game! The Proof Game! Here’s your chance to play the game that is quickly becoming a favorite among America’s teenagers: The Proof Game!

17. Example: Using CPCTC Given: ∠ ABD = ∠ CBD, ∠ ADB = ∠ CDB Prove: AB = CB A B C D ∠ ABD = ∠ CBD, ∠ ADB = ∠ CDB Given BD = BDReflexive Property ΔABD = ΔCBDASA (Angle-Side-Angle) AB = CBCPCTC (Corresponding Parts of Congruent Triangles are Congruent) Statement Reason

18. Your Turn: Using CPCTC Given: MO = RE, ME = RO Prove: ∠ M = ∠ R O R E M MO = RE, ME = ROGiven OE = OEReflexive Property ΔMEO = ΔROE SSS (Side-Side-Side) ∠ M = ∠ R CPCTC (Corresponding Parts of Congruent Triangles are Congruent) StatementReason

19. Your Turn: Using CPCTC Given: SP = OP, ∠ SPT = ∠ OPT Prove: ∠ S = ∠ O S P O T SP = OP, ∠ SPT = ∠ OPT Given PT = PTReflexive Property ΔSPT = ΔOPT SAS (Side-Angle-Side) ∠ S = ∠ O CPCTC (Corresponding Parts of Congruent Triangles are Congruent) ReasonStatement

20. Your Turn: Using CPCTC Given: KN = LN, PN = MN Prove: KP = LM K N L M P KN = LN, PN = MNGiven ∠ KNP = ∠ LNM Vertical Angles ΔKNP = ΔLNMSAS (Side-Angle-Side) KP = LM CPCTC (Corresponding Parts of Congruent Triangles are Congruent) StatementReason

21. Your Turn: Using CPCTC Given: ∠ C = ∠ R, ∠ T = ∠ P, TY = PY Prove: CT = RP C Y R P T ∠ C = ∠ R, ∠ T = ∠ P, TY = PY Given ΔTCY = ΔPRY AAS (Angle-Angle-Side) CT = RP CPCTC (Corresponding Parts of Congruent Triangles are Congruent) ReasonStatement

22. Your Turn: Using CPCTC Given: AT = RM, AT || RM Prove: ∠ AMT = ∠ RTM A T RM AT = RM, AT || RM Given ∠ ATM = ∠ RMT Alternate Interior Angles TM = TM Reflexive Property ΔTMA = ΔMTR SAS (Side-Angle-Side) ∠ AMT = ∠ RTM CPCTC (Corresponding Parts of Congruent Triangles are Congruent) StatementReason

23. Example 2 Visualize Overlapping Triangles SOLUTION Sketch the triangles separately and mark any given information. Think of ∆JGH moving to the left and ∆KHG moving to the right. 1. Sketch the overlapping triangles separately. Mark all congruent angles and sides. Then tell what theorem or postulate you can use to show ∆JGH  ∆KHG. Mark  GJH   HKG and  JHG   KGH.

24. Example 2 Visualize Overlapping Triangles Look at the original diagram for shared sides, shared angles, or any other information you can conclude. 2. Add congruence marks to GH in each triangle. In the original diagram, GH and HG are the same side, so GH  HG. You can use the AAS Congruence Theorem to show that ∆JGH  ∆KHG. 3.

25. Example 3 Use Overlapping Triangles In the original diagram,  C is the same in both triangles (  BCA   ECD). SOLUTION Sketch the triangles separately. Then label the given information and any other information you can conclude from the diagram. 1. Write a proof that shows AB  DE.  ABC   DEC AB  DE CB  CE

26. Show ∆ABC  ∆DEC to prove that AB  DE. Statements Reasons 1.  ABC   DEC Given1. Example 3 Use Overlapping Triangles Given2. CB  CE  C   C 3. Reflexive Prop. of Congruence 3. ASA Congruence Postulate4. ∆ABC  ∆DEC 4. Corresponding parts of congruent triangles are congruent. 5. AB  DE

27. Your Turn: Use Overlapping Triangles ANSWER SAS. 1. Tell which triangle congruence theorem or postulate you would use to show that AB  CD.

28. Your Turn: Use Overlapping Triangles ANSWER Statements Reasons 1.Given1. Given2.  J   L ASA Congruence Postulate4. ∆KJN  ∆KLM 4. Corresponding parts of  triangles are . 5. NJ  ML KJ  KL Reflexive Prop. of Congruence3.  K   K 2. Given KJ  KL and  J   L, show NJ  ML. Redraw the triangles separately and label all congruences. Explain how to show that the triangles or corresponding parts are congruent.

29. Your Turn: Use Overlapping Triangles Given  SPR   QRP and  Q   S, show ∆PQR  ∆RSP. 3. ANSWER Statements Reasons 1.Given1. Given2.  Q   S AAS Congruence Theorem4. ∆PQR  ∆RSP 4.  SPR   QRP Reflexive Prop. of Congruence3. PR  RP

30. Joke Time What happened to the man who lost the whole left side of his body? He is all right now. What do you find in an empty nose? Finger prints. What did one eye say to the other eye? Between you and me something smells.

31. Assignment Pg. 268 – 271:#1 – 29 odd