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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 Corresponding Parts Review: Congruence Shortcuts **Right triangles only: hypotenuse-leg (HL) 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 Example: Name the Congruence Shortcut or CBD SAS SSA CBD ASA SSS Name the Congruence Shortcut or CBD Reflexive Property SAS Vertical Angles SAS Vertical Angles SAS Reflexive Property SSA CBD Your Turn: Name the Congruence Shortcut or CBD Your Turn: Name the Congruence Shortcut or CBD Your Turn: Name the Congruence Shortcut or CBD Example Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: B For SAS: AC For AAS: A Your Turn: Indicate the additional information needed to enable us to apply the specified congruence postulate. For ASA: For SAS: For AAS: 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* Example 1 Use Corresponding Parts In the diagram, AB and CD bisect each other at M. Prove that A B. Example 1 Use Corresponding Parts Statements Reasons 1. AB and CD bisect each other at M. 1. Given 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. The Proof Game! Here’s your chance to play the game that is quickly becoming a favorite among America’s teenagers: The Proof Game! Rules: 1. Guys vs. Gals 2. Teams must take turns filling in the statements and reasons in the proofs to come. 3. If the statement/reason combo is correct, team gets 1 point. Next team continues. 4. If the statement/reason combo is incorrect, team loses 1 point. Next team fixes mistake. 5. Teammates cannot help the person at the board…he/she is on their own. Cheating loses all points!! Number One Given: ∠ABD = ∠CBD, ∠ADB = ∠CDB B Prove: AB = CB Statement Reason A C D Number Two Given: MO = RE, ME = RO Prove: ∠M = ∠R O Statement Reason M R E Number Three Given: SP = OP, ∠SPT = ∠OPT T S Prove: ∠S = ∠O Statement Reason P O Number Four Given: KN = LN, PN = MN Prove: KP = LM K L N Statement Reason P M Number Five Given: ∠C = ∠R, TY = PY Prove: CT = RP C R Y Statement Reason T P Number Six Given: AT = RM, AT || RM Prove: ∠AMT = ∠RTM A Statement T Reason M R Example 2 Visualize Overlapping Triangles 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 ∆JGH moving to the left and ∆KHG moving to the right. Mark GJH HKG and JHG KGH. Example 2 Visualize Overlapping Triangles 2. Look at the original diagram for shared sides, shared angles, or any other information you can conclude. In the original diagram, GH and HG are the same side, so GH HG. Add congruence marks to GH in each triangle. 3. You can use the AAS Congruence Theorem to show that ∆JGH ∆KHG. Example 3 Use Overlapping Triangles Write a proof that shows AB DE. ABC DEC CB CE AB DE SOLUTION Your Turn: Use Overlapping Triangles Redraw the triangles separately and label all congruences. Explain how to show that the triangles or corresponding parts are congruent. Given KJ KL and J L, show NJ ML. Your Turn: Use Overlapping Triangles 3. Given SPR QRP and Q S, show ∆PQR ∆RSP. 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. 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!!!