AMSCO'S Geometry. New York - Rye High School

358 Parallel LinesGivenProveABC with A B.CA CBCProofWe can use either the angle bisector or the altitude fromC to separate the triangle into two congruent triangles. Wewill use the angle bisector.ADBStatementsReasons1. Draw CD,the bisector of 1. Every angle has one and only oneACB.bisector.2. ACD BCD 2. An angle bisector of a triangle is aline segment that bisects an angleof the triangle.3. A B 3. Given.4. CD CD4. Reflexive property of congruence.5. ACD BCD 5. AAS.6. CA CB6. Corresponding parts of congruenttriangles are congruent.The statement of the Isosceles Triangle Theorem (Theorem 5.1) and its converse(Theorem 9.14) can now be written in biconditional form: Two angles of a triangle are congruent if and only if the sides opposite theseangles are congruent.To prove that a triangle is isosceles, we may now prove that either of the followingtwo statements is true:1. Two sides of the triangle are congruent.2. Two angles of the triangle are congruent.Corollary 9.13aIf a triangle is equiangular, then it is equilateral.GivenProveABC with A B C.ABC is equilateral.