GEOMETRY MODULE 5 LESSON 13 THE INSCRIBED ANGLE ALTERNATE – A TANGENT ANGLE OPENING EXERCISE Complete the Opening Exercise on page 97 of your workbook. SUMMARY OF LESSON 12 Two tangent segments to a If a circle is tangent to both Every triangle contains an circle from an exterior point rays of an angle, then its inscribed circle whose center are congruent. center lies on the angle is the intersection of the 퐽퐻̅̅̅̅ ≅ 퐹퐻̅̅̅̅ bisector. triangle’s angle bisector. _____________________________________________________________________________ Tangent-Secant Theorem: Let A be a point on a circle, let 퐴퐵⃗⃗⃗⃗⃗ be a tangent ray to the circle, and let C be a point on the circle such that 퐴퐶⃡⃗⃗⃗⃗ is a secant to the circle. If 푎 = 푚∠퐵퐴퐶 and b is the angle measure of the arc intercepted by 1 ∠퐵퐴퐶, then 푎 = 푏. 2 Theorem: Suppose 퐴퐵̅̅̅̅ is a chord of circle C, and 퐴퐷̅̅̅̅ is tangent to the circle at point A. If E is any point other than A or B in the arc of C on the opposite side of 퐴퐵̅̅̅̅ from D, then 푚∠퐵퐸퐴 = 푚∠퐵퐴퐷. NOTE: ∠퐵퐸퐴 is the angle made by chords opposite 퐴퐵̅̅̅̅. ∠퐵퐴퐷 is the angle made by chord 퐴퐵̅̅̅̅ and the tangent segment 퐴퐷̅̅̅̅. 1 MOD5 L13 PRACTICE Together, we will complete exercise 1 and 2 on page 101 of your workbook. Chord 퐵퐶̅̅̅̅ and Tangent 퐵퐺̅̅̅̅ make a 34° angle. D is a point on the opposite side of chord 퐵퐶̅̅̅̅. So, 푚∠푎 = 푚∠퐶퐵퐺. 푎 = 34° By Thales, 푚∠퐸퐷퐵 = 90° 푎 + 푏 = 90° 푏 = 90° − 34° 푏 = 56° ∠푐 and ∠퐷퐶퐵 are both inscribed angles and they share arc 퐷퐵̂ . 푚∠푐 = 푚∠퐷퐶퐵 푐 = 52° The angle supplementary to ∠퐴퐵퐷 equals 74°. By the Tangent Secant Theorem, 푏 = 2(74) 푏 = 148° Triangle ABC is isosceles. 180 − 148 푎 = 2 푎 = 16° 2 MOD5 L13 ON YOUR OWN Complete exercise 3 on page 102 of your workbook. The angle supplementary to ∠퐶퐵퐷 equals 43°. By the Tangent Secant Theorem, 푎 = 2(43) 푎 = 86° ∠푏 is an inscribed angle and shares the intercepted arc with central angle ∠푎 1 푚∠푏 = 푚∠푎 2 푏 = 43° OR Chord 퐵퐷̅̅̅̅ and Tangent 퐶퐵̅̅̅̅ make a 43° angle. E is a point on the opposite side of chord 퐵퐷̅̅̅̅. So, 푚∠푏 equal the tangent angle. 푏 = 43° HOMEWORK Problem Set Module 5 Lesson 13, page 104. #1, #2, #3, #4, and #5. You must show all calculations and/or state an explanation! DUE: Monday, May 8, 2017 3 MOD5 L13 .