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Geometry Honors Section 9.3 Arcs and Inscribed Angles Recall that a *central angle is an angle whose vertex is at the center of the circle and whose sides are radii. What is the relationship between a central angle and the arc that it cuts off? The measure of the central angle equals the measure of its intercepted arc. An *inscribed angle is an angle whose vertex lies on the circle and whose sides are chords. A E T By doing the following activity, you will be able to determine the relationship between the measure of an inscribed angle and the measure of its intercepted arc. Given the measure of 1, complete the table. Remember that the radii of a circle are congruent. 0 0 0 20 40 40 0 0 0 30 60 60 0 0 0 x 2x 2x What does the table show about the relationship between m1 and mPK ? 1 m1 mPK 2 Inscribed Angle Theorem The measure of an angle inscribed in a circle is equal to ½ its intercepted arc. 35 0 35 0 90 0 700 Corollaries of the Inscribed Angle Theorem: If two inscribed angles intercept the same arc, then the angles are congruent. If an inscribed angle intercepts a semicircle, then the angle is a right angle. 0 65 0 65 1300 500 0 35 0 35 50 0 0 90 0 120 700 1100 A second type of angle that has its vertex on the circle is an angle formed by a tangent and a chord intersecting at the point of tangency. 0 0 120 30 0 0 100 40 0 0 80 50 90 0 90 0 90 0 0 60 0 50 0 40 Theorem: If a tangent and a chord intersect on a circle at the point of tangency, then the measure of the angle formed is equal to ½ the measure of the intercepted arc. 75 0 194 0 1660 830

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