AMSCO'S Geometry. New York - Rye High School

554 Geometry of the CircleCorollary 13.9aAn angle inscribed in a semicircle is a right angle.Proof: In the diagram, AOC is a diameter of circle O, andABC is inscribed in semicircle ABC. Also ADC is asemicircle whose degree measure is 180°. Therefore,mABC 5 1 2 mADC5 1 2 (180)5 90XXXADOBCSince any triangle can be inscribed in a circle, the hypotenuse of a trianglecan be the diameter of a circle with the midpoint of the hypotenuse the centerof the circle.Corollary 13.9bIf two inscribed angles of a circle intercept the same arc, then they arecongruent.Proof: In the diagram, ABC and ADC areinscribed angles and each angle intercepts1ACX . Therefore, mABC X2 and1mADC XmAC 2 mAC . Since ABC and ADChave equal measures, they are congruent.ABCDEXAMPLESolutionTriangle ABC is inscribed in circle O,mB 70, and mBCXa. mACX b. mA c. mC d. mABXa. If the measure of an inscribed angle is one-halfthe measure of its intercepted arc, then the measureof the intercepted arc is twice the measureof the inscribed angle.m/B 5 1 X22m/B 5 mACXmAC2(70) 5 mACX140 5 mACX 100. Find:B70A100C