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4.5 Proving Triangles are Congruent: ASA and AAS Hubarth Geometry Angle-Side-Angle Congruence Postulate (ASA) Words 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. Symbols If Angle A X, and Side AB XY , and Angle B Y, Then ABC XYZ Ex 1. Determine When to Use ASA Congruence Based on the diagrams can you use the ASA Congruence Postulate to show that the triangles are congruent? Explain your reasoning. a. B O M W A b. N Solution a. B M , O A, and BO AM , So by ASA triangleBOW AMN b. A C, ABB' CBB' and AB CB, So, by ASA triangle ABB' CBB'. Angle-Angle-Side Congruence Theorem (AAS) Words If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the two triangles are congruent. Symbols If Angle A O, and Angle N W, and Side BO MA, then BOW MAN Ex 2. Determine What Information Is Missing What additional congruence is needed to show that JKL NML by the AAS. K N L J M Solution Given: KL ML KLJ MLN by vertical angles To use AAS we will need to know J N. Ex 3. Decide Whether Triangles are Congruent Does the diagram give enough information to show that the triangles are congruent? If so, state the reason. N Q a. U H b. E Z c. G F J M W P X Solution a. EF HJ Given E J Given FGE HGJ Vertical Angles EFG JHG AAS b. MP QN Given c. WZ WZ Reflexive Prop NP NP Reflexive Prop. UZW XWZ Alt. Int No other information is given to determine that UWZ XZW Alt Int they are congruent. UZW XWZ ASA Ex 4. Prove Triangles are Congruent A step in the Cat's Cradle string game creates the triangles shown. Prove that ABD EBC. A C B D Solution Given: BD BC , AD EC Prove: ABD EBC Statements Reasons 1. BD BC 1. Given 2. AD EC 3. D C 2. Given 3. Alt. Int. angles 4. ABD EBC 4. Vertical Angles 5. ABD EBC 5. ASA E Practice 1. Based on the diagram, can you use the ASA Congruence Postulate to show the triangles are congruent? Explain. E D B A D, B E AB DE , ABC DEF by ASA F A C 2. Does the diagram give enough information to show that the triangles are congruent? If so state why. B A Yes, AAS D C 3. Use the information in the diagram to prove that FGH GJK. F G H G’ K J Statements 1. GFH JG'K Reasons 1. Given 2. GF JG ' 3. FGH G'JK 4. FGH G'JK 2. Given 3. Corresponding Angles 4. ASA