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N#____ ____/____/____ 4.6 & 4.7 Congruence in Right Triangles & Overlapping Triangles A#_____ _____/_____/_____ WS 4.6 & 4.7 Hypotenuse-Leg Theorem : If a hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle then the triangles are congruent. X Y M Z L N E1 Given : PS PT, and PRS PRT Prove : PRS PRT P S Statement PRS PRT PRS PRT are right angles. Reason given linear pair PRS & PRT are right 's. def of right triangles PS PT PR PR given reflexive Hypotenuse Leg PRS PRT R T What additional information would prove each pair of triangles congruent by Hypotenuse-Leg Theorem? E2 M E3 Q Z X D K N O R T P mZ mT 90 MO QR For what values of x or y are the triangles congruent by Hypotenuse-Leg Theorem? X Z 2y-10 E4 D K T 2 y 10 90 y 50 F A E5 15 B 3x+7 C E 15 3x 7 8 x 3 D CPCTC: Corresponding Parts of Congruent Triangles are Congruent. X Y M Z L N Separate and redraw the overlapping triangles. Identify the common triangles and angles or sides. E1 X E2 E Z V B C A E E Y X Z Z B A A AE AE V V C ZV ZV Y Separate and redraw the overlapping triangles. Identify the common triangles and angles or sides. E3 E4 X D A V E Z U W B X C Y A BC BC X Z V D W B X X C B C Y E5 Given : CA CE , C and BA DE Prove : BXA DXE Statement 1) BA DE 2) CA CE B X A Reason 1) given 2) given 3)CAE CEA 3) Isosceles triangletheroem 4) AE AE 4) reflexive 5) BAE DEA 6) ABE EDA 5) SAS 7) BXA DXE 7) vertical 8) BXA DXE 8) AAS 6) CPCTC D E