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Using Congruent Triangles

Using Congruent Triangles. Chapter 4. Objective. List corresponding parts. Prove triangles congruent (ASA, SAS, AAS, SSS, HL) Prove corresponding parts congruent (CPCTC) Examine overlapping triangles. Key Vocabulary - Review. Reflexive Property Vertical Angles Congruent Triangles

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Using Congruent Triangles

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  1. Using Congruent Triangles Chapter 4

  2. Objective • List corresponding parts. • Prove triangles congruent (ASA, SAS, AAS, SSS, HL) • Prove corresponding parts congruent (CPCTC) • Examine overlapping triangles.

  3. Key Vocabulary - Review • Reflexive Property • Vertical Angles • Congruent Triangles • Corresponding Parts

  4. Review: Congruence Shortcuts **Right triangles only: hypotenuse-leg (HL)

  5. Congruent Triangles (CPCTC) 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 SSA SSS CBD

  7. Name the Congruence Shortcut or CBD Vertical Angles Reflexive Property SAS SAS Reflexive Property Vertical Angles SSA SAS CBD

  8. Your Turn: Name the Congruence Shortcut or CBD

  9. Your Turn: Name the Congruence Shortcut or CBD

  10. Your Turn: Name the Congruence Shortcut or CBD

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

  12. Your Turn: Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS:

  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 In the diagram, AB and CD bisect each other at M. Prove that A B. Use Corresponding Parts

  15. Example 1 Statements Reasons 1. AB and CD bisect each other at M. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. Use Corresponding Parts 1. Given

  16. 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. Rules: Guys vs. Gals Teams must take turns filling in the statements and reasons in the proofs to come. If the statement/reason combo is correct, team gets 1 point. Next team continues. If the statement/reason combo is incorrect, team loses 1 point. Next team fixes mistake. Teammates cannot help the person at the board…he/she is on their own. Cheating loses all points!!

  18. Number One Given: ∠ABD = ∠CBD, ∠ADB = ∠CDB Prove: AB = CB B A C Statement Reason D

  19. Number Two Given: MO = RE, ME = RO Prove: ∠M = ∠R O R Statement Reason M E

  20. Number Three Given: SP = OP, ∠SPT = ∠OPT Prove:∠S = ∠O O T S Statement Reason P

  21. Number Four Given: KN = LN, PN = MN Prove: KP = LM K L N Statement Reason M P

  22. Number Five Given:∠C = ∠R, TY = PY Prove: CT = RP C R Y Statement Reason P T

  23. Number Six Given: AT = RM, AT || RM Prove:∠AMT = ∠RTM A T Statement Reason M R

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

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

  26. Example 3 Write a proof that shows ABDE. ABC DEC CB CE AB DE Use Overlapping Triangles SOLUTION

  27. Your Turn: Redraw the triangles separately and label all congruences. Explain how to show that the triangles or corresponding parts are congruent. GivenKJ KLandJ L,showNJML. Use Overlapping Triangles

  28. Your Turn: 3. Given SPR QRPand Q S, show ∆PQR  ∆RSP. Use Overlapping Triangles

  29. Joke Time • What happened to the man who lost the whole left side of his body? • He is all right now. • What did one eye say to the other eye? • Between you and me something smells.

  30. Upcoming Schedule • Quiz on Friday…HL, proofs, CPCTC, Isosceles Triangle Thm, overlapping triangles • Monday – vocabulary terms • Tues – Practice Day • Wednesday – Chapter 4 Test • **reminder – projects due Oct. 27!!!

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