AMSCO'S Geometry. New York - Rye High School

540 Geometry of the CircleTheorem 13.2aIn a circle or in congruent circles, if central angles are congruent, then theirintercepted arcs are congruentGiven Circle O circle O, AOB COD,and AOB AOB.Prove ABX > CD X and ABX > ArBr X.ACBOABOProofIt is given that AOB COD andDAOB AOB. Therefore,mAOB mCOD mAOB because congruent angles have equalmeasures. Then since the degree measure of an arc is equal to the degree measureof the central angle that intercepts that arc, mABX 5 mCD X 5 mArBr . It isalso given that circle O and circle O are congruent circles. Congruent arcs aredefined as arcs of the same circle or of congruent circles that are equal in measure.Therefore, since their measures are equal, ABX > CD X and ABX > ArBr .XXThe converse of this theorem can be proved by using the same definitionsand postulates.Theorem 13.2bIn a circle or in congruent circles, central angles are congruent if their interceptedarcs are congruent.Theorems 13.2a and 13.2b can be written as a biconditional.Theorem 13.2In a circle or in congruent circles, central angles are congruent if and only iftheir intercepted arcs are congruent.EXAMPLE 1Let OA hand OBhbe opposite rays and mAOC 75. Find:a. mBOC b. mACXc. mBCXd. mABX e. mBAC XSolution a. m/BOC 5 m/AOB 2 m/AOC5 180 2 755 105DA75°OBC