11.2.1 Inscribed Angles (continued)
Example 2 Recreation Application
If inscribed angles of a circle intercept the same arc or are subtended by the same chord, then the angles are congruent. Match up the inscribed angles that intercept the same arcs in the diagram with those angles listed as congruent. The measure of an inscribed angle is determined in this application example. m ∠ PTQ is given as 50 ° and RS m is given as 110 ° . It is possible to use the given information to find the measures of angles in the triangle that contains ∠ QZR . According to the Inscribed Angle Theorem, = PQ m 100 and therefore m ∠ QRZ = 50 ° . Also according to the Inscribed Angle Theorem, m ∠ RQZ = 55 ° . Knowing two of the three angles in Δ QRZ , the Triangle Sum Theorem can be used to find the measure of the unknown angle. Thus, m ∠ QZR = 75 ° .
Example 3 Finding Angle Measures in Inscribed Triangles
An inscribed angle subtends a semicircle if and only if the angle is a right angle. To subtend something means to be opposite it and to extend from one endpoint to the other endpoint. Think about the measure of the right angle and the measure of the arc, a semicircle, formed by the right angle. Is this theorem a special case of the Inscribed Angles Theorem?
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