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### Understanding Inscribed Angles and Their Theorems ###

This document explores the Inscribed Angle Theorem, which states that the measure of an inscribed angle is half that of its intercepted arc. It also covers important corollaries, such as the congruence of inscribed angles that intercept the same arc, the fact that an angle inscribed in a semicircle is a right angle, and the supplementary nature of opposite angles in a quadrilateral inscribed in a circle. Additionally, we analyze problems involving these theorems to reinforce understanding. ###

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### Understanding Inscribed Angles and Their Theorems ###

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  1. 11-3Inscribed Angles Theorems: Inscribed Angle Theorem, 11-10 Corollaries: Corollaries to the Inscribed Angle Theorem Vocabulary: Inscribed angle, Intercepted arc

  2. Vocabulary A Intercepted Arc Inscribed Angle C B

  3. Theorem 11-9 (Inscribed Angle Theorem) • The measure of an inscribed angle is half the measure of its intercepted arc. A B C

  4. #1 Using the Inscribed Angle Theorem P ao 60o T Q 30o S bo 60o R

  5. Corollaries to the Inscribed Angle Theorem • Two inscribed angles that intercept the same arc are congruent. • An angle inscribed in a semicircle is a right angle. • The opposite angles of a quadrilateral inscribed in a circle are supplementary.

  6. #2 Using Corollaries to Find Angle Theorem • Find the diagram at the right, find the measure of each numbered angle. 120o 1 60o 2 4 80o 3 100o

  7. Theorem 11-10 • The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. B B D D C C

  8. #3 Using Theorem 11-10 • Find x and y. J 90o Q 35o yo 55o xo L K

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