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104 Use Inscribed Angles and Polygons 1 Inscribed Angle Vertex lies on the circle, the sides are the chords. A C Example: <ABC is an inscribed angle of circle P. P B What is <APC? Theorem 107 Measure of an Inscribed Angle Theorem The measure of an inscribed angle is one half the measure of its intercepted arc. Example: If <ABC = 350, then A C mAC = ______ mABC = _______ P B What type of angle is <APC? ____________ What is the m<APC? ___________ http://www.ronblond.com/M11/INSC.INSC.APPLET/index.html 2 Theorem 108 If 2 inscribed angles of a circle intercept the same arc, then the angles are congruent. A D <ADB ≅ <ACB C B Examples: 1. In a regular inscribed 5 point star. Find the measure of each inscribed angle. 3 2.) A B m<ACB = 300 mAB = ________ C D m<ADB = _______ 4 Theorem 109 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if 1 side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. A m<ABCD = 900 iff AC is a diameter of the circle. D B C Theorem 1012 A quadrilateral can be inscribed in a circle iff its opposite angles are supplementary. Can you inscribe a 1. Square? 1. yes 2. Parallelogram? 2. No 3. Rectangle? 3. Yes 4. Trapezoid? 5. Isosceles Trapezoid? 6. Kite? 4. No 5. Yes 6. No The above is a circumscribed circle and inscribed quadrilateral! 5