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Transcript
Geometry 4-5 Proving Triangles Congruent - ASA, AAS Write a flow proof. An included side is the side of a polygon located between two consecutive angles. Given: ZX bisects WZY Z and YXW Prove: WXZ~ = XZY ZX bisects WZY ZX bisects WXY Given ~ YZX WZX= WXZ~ = YXZ Def'n bisector ZX ~ = ZX Reflexive WXZ ~ = XZY ASA Postulate 4.3 - Angle-Side-Angle (ASA) Congruence: 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. W X Y R S Theorem 4.5 - Angle-Angle-Side (AAS) Theorem: If two angles and the nonincluded side of one triangle are congruent with two angles and the nonincluded side of a second triangle, then the two triangles are congruent. ~ ST and RQ || ST Given: RQ = Prove: RUQ ~ = TUS There are a few congruence theorems that apply ONLY to right triangles. Theorem 4.8 - Leg-Angle (LA) Congruence: If one leg and an acute angle of one right triangle are congruent to one leg and an acute angle of a second triangle, then the two triangles are congruent. (Basically ASA) Theorem 4.6 - Leg-Leg (LL) Congruence: If the legs of one right triangle are congruent to the legs of a second right triangle, then the triangles are congruent. (Basically SAS) Theorem 4.7 - Hypotenuse-Angle (HA) Congruence: If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and an acute angle of a second right triangle, then the two triangles are congruent. (Basically AAS) Statements 1. RQ || ST 2. RQU~ = UST 3. RUQ ~ = TUS ~ ST 4. RQ = 5. RUQ ~ = TUS U Reasons 1. Given Q T 2. Alt Int 's Thm 3. Vert 's Thm 4. Given 5. AAS Theorem 4.9 - Hypotenuse-Leg (HL) Congruence: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent. (Basically SSS)