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Geometry 4-3 Congruent Triangles If two or more geometric figures are exactly the same size and the same shape, they are called congruent . In congruent polygons, all of the parts of one polygon are congruent to the corresponding parts of the other polygon. Show that the triangles are congruent by identifying all congruent corresponding parts. Then write a congruence statement. P X X~ = Z P XY ~ = PQ XYZ ~ = Given ABC~ = DFE, find the values of x and y. D B 54 45 2y + x 32.1 99 36 A 8y - 5 F C 38.4 E 8y - 5 = 99 8y = 104 y = 13 2y + x = 54 2(13) + x = 54 26 + x = 54 x = 28 Theorem 4.4 - Properties of Triangle Congruence: The Reflexive, Symmetric, and Transitive Properties of Equality also hold true for congruence. ABC ~ = ABC If ABC~ = EFG, then EFG~ = ABC. If ABC ~ = EFG and EFG~ = JKL, then ABC~ = JKL. Y Y~ = R Q R~ = Z YZ ~ = QR RP ~ = ZX Q PQR Theorem 4.3 - Third Angles Theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the two triangles are also congruent. G E Given: DE~ =GE, DF~ =GF, D~ = G, DFE~ = GFE Prove: DEF~ = GEF D F Statements Reasons 1. DE~ 1. Given =GE, DF~ =GF, D~ = G, DFE~ = GFE 2. Reflexive 2. EF ~ = EF 3. DEF~ 3. Third Angle Theorem = GEF 4. DEF~ 4. Def'n ~ = GEF = 's