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Geometry Lesson 4.4 Proving Triangles are Congruent: ASA and AAS Warm Up: Review of SSS / SAS Write the method you would use to show the two triangles are congruent based on the information given in the diagram and any other information you can determine (a) (c) SSS SSS (b) (d) SAS SAS 1. Angle-Side-Angle (ASA) If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent AC is the included side of A & C B A C D E Angle: A D Side: AC DF Angle: C F ABC DEF Order is F Important!! 2. Angle-Angle-Side (AAS) If two angles and a non-included side of one triangle are congruent to two angles and a non-included side of a second triangle, then the triangles are congruent BC is a nonincluded side of A & C B C A D E Angle: A D Angle: C F Side: BC EF ABC DEF F Order is Important!! Example 1a/b: Can You Prove It? Is it possible to prove the s are ? How? E H N Q G F J Know: EF JH & E J Can show: EGF HGJ (Vertical s) EGF JGH by AAS M P Know: NQ PM Can show: NP NP (Reflexive) Not enough info. What do you know about the figures? What can you show to help show congruence? Example 1c: Can you Prove It? Is it possible to prove the s are ? How? U 4 W 3 1 X Z 2 Know: UW || XZ Can show: 1 3 2 4 (alt. int. s) WZ WZ (reflexive) UZW XWZ by ASA What do you know about the figures? What can you show to help show congruence? Practice 1 Is it possible to prove the s are ? How? C H E I D G F Know: C F & G E Can show: CDG FDE (vertical s) AAA not enough K J Know: KHJ IJH KJH IHJ Can show: HJ HJ KHJ IJH by ASA What do you know about the figures? What can you show to help show congruence? Example 2: What is Needed? What information do you need to show the triangles are using the specified method? (a) (b) ASA AAS C F C F Practice 2: What is Needed? What information do you need to show the triangles are using the specified method? N (a) E (b) AAS F SAS D M DF MO O E N F D M D M O Example 3a: Creating a Proof C Given: AD || EC; BD BC A Prove: ABD EBC Plan for proof: We have two sides and two || lines. Our options are SSS, SAS, ASA, or AAS. What do we need? Statement Reason B D AD || EC & BD BC Given ABD EBC Vertical Angles are D C Alt. Int. Angles are ABD EBC ASA E Example 3b: Creating a Proof Given: B C, D F, M is the midpoint of DF Prove: BDM CFM B C Plan for proof: We have two angles and a midpoint. Our options are ASA or AAS. What’s needed? D Statement Reason B C, D F M is midpoint of DF DM FM BDM CFM M Given Given Definition of a midpoint AAS F Practice 3a: Creating a Proof Given: LA || SN, LR NR Prove: LAR NSR Plan for proof: SSS? SAS? ASA? AAS? Statement LA || SN, LR NR LRA NRS ALR SNR LAR NSR Reason Given Definition of vertical angles alt. int. s are ASA Practice 3b: Creating a Proof Given: AC || BD, AB || CD Prove: ABC DCB Plan for proof: Statement Reason AC || BD, AB || CD ACB DBC Given Alt. int. s ABC DCB BC BC ABC DCB Alt. int. s SSS? SAS? ASA? AAS? Reflexive Property ASA Assignment 4.4 (pg. 223-224) #8-22 EVEN