11.2.1 Inscribed Angles Key Objectives • Find the measure of an inscribed angle. • Use inscribed angles and their properties to solve problems. Key Terms
• An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. • An intercepted arc consists of endpoints that lie on the sides of an inscribed angle and all the points of the circle between them. • A chord or arc subtends an angle if its endpoints lie on the sides of the angle. Theorems, Postulates, Corollaries, and Properties • Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc. ∠ = ABC AC m m • Corollary If inscribed angles of a circle intercept the same arc or are subtended by the same chord or arc, then the angles are congruent. • Theorem An inscribed angle subtends a semicircle if and only if the angle is a right angle. • Theorem If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. Example 1 Finding Measures of Arcs and Inscribed Angles 1 2
According to the Inscribed Angle Theorem, the measure of an inscribed angle is half the measure of its intercepted arc. Notice how this theorem covers three different cases of inscribed angles. Also, think about how this differs from the relationship of a central angle and its arc. The measure of an inscribed angle and the measure of an arc intercepted by an inscribed angle are determined in these examples. In the first example, the measure of the arc intercepted by the inscribed angle CDB is given. According to the Inscribed Angle Theorem, the measure of the inscribed angle is equal to one-half the measure of the arc, ∠ = = = CDB CB m (1/2)m (1/2)80 40 . In the second example, the measure of the inscribed angle that intercepts the arc is given. According to the Inscribed Angle Theorem, the measure of the arc is twice the measure of the inscribed angle that intercepts it, = ∠ = = AD ACB m 2m 2(50 ) 100 .
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