AMSCO'S Geometry. New York - Rye High School

The Converse of the Isosceles Triangle Theorem 359ProofWe are given equiangular ABC. Then since A B, the sides oppositethese angles are congruent, that is, BC > AC. Also, since B C,AC > AB for the same reason. Therefore, AC > AB by the transitive propertyof congruence, AB > BC > CA, and ABC is equilateral.EXAMPLE 1In PQR, Q R. If PQ 6x 7 and PR 3x 11, find:a. the value of x b. PQ c. PRSolutionEXAMPLE 2a. Since two angles of PQRare congruent, the sidesopposite these angles arecongruent. Thus, PQ PR.6x 7 3x 116x 3x 11 73x 18x 6 Answerb. PQ 5 6x 2 7 c. PR 5 3x 1 115 6(6) 2 75 3(6) 1 115 36 2 75 18 1 115 29 Answer 5 29 AnswerThe degree measures of the three angles of ABC are represented bymA x 30, mB 3x, and mC 4x 30. Describe the triangle as acute,right, or obtuse, and as scalene, isosceles, or equilateral.Solution The sum of the degree measures of the angles of a triangle is 180.x 30 3x 4x 30 1808x 60 1808x 120x 15Substitute x 15 in the representations given for the three angle measures.m/A 5 x 1 305 15 1 305 45m/B 5 3x5 3(15)5 45m/C 5 4x 1 305 4(15) 1 305 60 1 305 90Since A and B each measure 45°, the triangle has two congruent anglesand therefore two congruent sides. The triangle is isosceles. Also, since oneangle measures 90°, the triangle is a right triangle.AnswerABC is an isosceles right triangle.