10.4 Warm Up Warm Up Lesson Quiz Lesson Quiz Lesson Presentation Lesson Presentation Use Inscribed Angles and Polygons

10.4 Warm-Up ANSWER The measure of the interior angles of a quadrilateral are 80º, 100º, 55º, and 5xº. Find the value of x. 2. Two supplementary angles have measures 6xº and 12xº. Find each angle measure. ANSWER 60º; 120º

10.4 Warm-Up ANSWER Solve 3x = ( 4x + 12) Solve 80 = ( 360 – 2x). 1212

10.4 Example 1 a. m T mQRb. Find the indicated measure in P. SOLUTION M T = mRS = (48 o ) = 24 o a. mQR = 180 o mTQ = 180 o 100 o = 80 o. So, mQR = 80 o. – – mTQ = 2m R = 2 50 o = 100 o. Because TQR is a semicircle, b.

10.4 Example 2 Find mRS and m STR. What do you notice about STR and RUS ? SOLUTION From Theorem 10.7, you know that mRS = 2m RUS = 2 (31 o ) = 62 o. Also, m STR = mRS = (62 o ) = 31 o. So, STR RUS

10.4 Example 3 SOLUTION Notice that JKM and JLM intercept the same arc, and so JKM JLM by Theorem Also, KJL and KML intercept the same arc, so they must also be congruent. Only choice C contains both pairs of angles. So, by Theorem 10.8, the correct answer is C.

10.4 Guided Practice Find the measure of the red arc or angle. 1. ANSWER 45°

10.4 Guided Practice Find the measure of the red arc or angle. ANSWER 76° 2.

10.4 Guided Practice Find the measure of the red arc or angle. ANSWER 72° 3.

10.4 Example 4 Photography Your camera has a 90° field of vision and you want to photograph the front of a statue. You move to a spot where the statue is the only thing captured in your picture, as shown. You want to change your position. Where else can you stand so that the statue is perfectly framed in this way?

10.4 Example 4 SOLUTION From Theorem 10.9, you know that if a right triangle is inscribed in a circle, then the hypotenuse of the triangle is a diameter of the circle. So, draw the circle that has the front of the statue as a diameter. The statue fits perfectly within your camera’s 90 o field of vision from any point on the semicircle in front of the statue.

10.4 Guided Practice 4. WHAT IF? In Example 4, explain how to find locations if you want to frame the front and left side of the statue in your picture. Make the diameter of your circle the diagonal of the rectangular base. ANSWER

10.4 Example 5 Find the value of each variable. a. SOLUTION PQRS is inscribed in a circle, so opposite angles are supplementary. a. m P + m R = 180° 75° + y° = 180° y = 105 m Q + m S = 180° 80° + x° = 180° x = 100

10.4 Example 5 Find the value of each variable. b. JKLM is inscribed in a circle, so opposite angles are supplementary. m J + m L = 180° 2a° + 2a° = 180° a = 45 m K + m M = 180° 4b° + 2b° = 180° b = 30 4a = 1806b = 180 b. SOLUTION

10.4 Guided Practice 5. Find the value of each variable. x = 98, y = 112 ANSWER

10.4 Guided Practice Find the value of each variable. c = 62, x = 10 ANSWER 6.

10.4 Lesson Quiz ANSWER 38 Find the value of x ANSWER 56

10.4 Lesson Quiz ANSWER 44 Find the value of x. 3.

10.4 Lesson Quiz 4. Find the value of each variable. ANSWER x = 54; y = 20

10.4 Lesson Quiz 5. Find the value of each variable. ANSWER x = 5; y = 10