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Name_____________________________________ Class____________________________ Date________________ Lesson 12-3 Inscribed Angles Lesson Objectives 1 Find the measure of an inscribed angle 2 Find the measure of an angle formed by a tangent and a chord NAEP 2005 Strand: Geometry Topic: Relationships Among Geometric Figures Local Standards: ____________________________________ Vocabulary and Key Concepts. All rights reserved. Theorem 12-9: Inscribed Angle Theorem A The measure of an inscribed angle is B C mB Theorem 12-10 The measure of an angle formed by a tangent and a chord is B B D D C C Corollaries to the Inscribed Angle Theorem 1. Two inscribed angles that intercept the same arc are 2. An angle inscribed in a semicircle is a . angle. 3. The opposite angles of a quadrilateral inscribed in a circle are An inscribed angle has . A B C An intercepted arc is 234 Geometry Lesson 12-3 Daily Notetaking Guide © Pearson Education, Inc., publishing as Pearson Prentice Hall. mC Name_____________________________________ Class____________________________ Date ________________ Examples. x 1 mDEF 70° E 1 Using the Inscribed Angle Theorem Find the values of x and y. F 80° Inscribed Angle Theorem C x (m ) m D y° Arc Addition Postulate x° © Pearson Education, Inc., publishing as Pearson Prentice Hall. All rights reserved. x ( ) 90° Substitute. x G Simplify. 0 1 Because EFG is the intercepted arc of D, you need to find mFG in order 1 to find mEFG . The arc measure of a circle is 360, so 0 . mFG 360 y 1 mEFG y (m y ( Inscribed Angle Theorem m ) Arc Addition Postulate ) Substitute. y Simplify. 2 Using Corollaries to Find Angle Measures Find the values of a and b. By Corollary 2 to the Inscribed Angle Theorem, an angle inscribed in a semicircle is a right angle, so a . b° 32° O The sum of the measures of the three angles of the triangle inscribed in O is . Therefore, the angle whose intercepted arc has a° measure b must have measure 180 Because the inscribed angle has the measure of the intercepted arc, the intercepted arc has angle, so b 2( Daily Notetaking Guide ) or . the measure of the inscribed . Geometry Lesson 12-3 235 Name_____________________________________ Class____________________________ Date________________ B 3 *Using ) Theorem 12-10 RS and TU are diameters of circle A. T RB is tangent to A at point R. Find mBRT and mTRS. 0 mRT m mBRT Theorem 12-10 m N Arc Addition Postulate ( mBRT ) Substitute and 126° A S for mURT for mUR. U Simplify. All rights reserved. mBRT R 0 mRT Use the properties of tangents to find mTRS. A tangent is mBRS the radius of a circle at its point of tangency. mBRS m to m mTRS mTRS Angle Addition Postulate Substitute. Simplify. Quick Check. 2. For the diagram at the right, find the measure of each numbered angle. 4 1 60° 2 80° 3 3. In Example 3, describe two ways to find mNRS using Theorem 12-10. 236 Geometry Lesson 12-3 Daily Notetaking Guide © Pearson Education, Inc., publishing as Pearson Prentice Hall. 1. In Example 1, find mDEF.