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Geometry January 4, 2011 Take out: Pencil, Homework, binder and begin the Do Now silently! Objectives: SWBAT: Describe what central and inscribed angles are Apply the relationship between the angle and its intercepted arc to find missing angle and arc measurements Agenda Do Now (10 min) Angles in circles ppt (10 min) Practice using conjectures (30 min) Angles in a Circle 2 Central Angle Definition: An angle whose vertex lies on the center of the circle. Central Angle (of a circle) Central Angle (of a circle) NOT A Central Angle (of a circle) 3 Central Angle Theorem The measure of a central angle is equal to the measure of the intercepted arc. Intercepted Arc Central Angle O Y 110 Z 4 Central Angle Theorem The measure of a central angle is equal to the measure of the intercepted arc. Example: Give AD is the diameter, find the value of x and y and z in the figure. B 25 A C x y O 55 z D x 25 y 180 (25 55 ) 180 80 100 z 55 5 Inscribed Angle Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle). ABC is an inscribed angle. No! B O Examples: 1 C A D 3 2 Yes! No! 4 Yes! 6 Intercepted Arc Intercepted Arc: An angle intercepts an arc if and only if each of the following conditions holds: 1. The endpoints of the arc lie on the angle. 2. All points of the arc, except the endpoints, are in the interior of the angle. 3. Each side of the angle contains an endpoint of the arc. C B ADC is the int ercepted arc of ABC. O A D 7 Inscribed Angle Theorem The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Y Inscribed Angle A 55 C D Z Intercepted Arc B An angle formed by a chord and a tangent can be considered an inscribed angle. mAB mABC 2 8 An angle inscribed in a semicircle is a right angle. P S 180 90 R 9