1. The measure of the interior angles of a quadrilateral are 80º, 100º, 55º, and 5xº. Find the value of x. ANSWER 25 2. Two supplementary angles have measures 6xº and 12xº. Find each angle measure. ANSWER 60º; 120º

3. Solve 3x = ( 4x + 12). 1 2 ANSWER 6 4. Solve 80 = ( 360 – 2x). 1 2 ANSWER 100

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

EXAMPLE 2 Find the measure of an intercepted arc 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 (31o) = 62o. Also, m STR = mRS = (62o) = 31o. So, STR RUS. 1 2

EXAMPLE 3 Standardized Test Practice SOLUTION Notice that JKM and JLM intercept the same arc, and so JKM JLM by Theorem 10.8. 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.

GUIDED PRACTICE for Examples 1, 2 and 3 Find the measure of the red arc or angle. 1. SOLUTION m G = mHF = (90o) = 45o 1 2 a.

GUIDED PRACTICE for Examples 1, 2 and 3 Find the measure of the red arc or angle. 2. SOLUTION mTV = 2m U = 2 38o = 76o. b.

GUIDED PRACTICE for Examples 1, 2 and 3 Find the measure of the red arc or angle. 3. SOLUTION Notice that ZYN and ZXN intercept the same arc, and so ZYN by Theorem 10.8. Also, KJL and KML intercept the same arc, so they must also be congruent. ZXN ZYN ZXN ZXN 72°

EXAMPLE 4 Use a circumscribed circle Photography Your camera has a 90o 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?

EXAMPLE 4 Use a circumscribed circle 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 90o field of vision from any point on the semicircle in front of the statue.

GUIDED PRACTICE for Example 4 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. SOLUTION Make the diameter of your circle the diagonal of the rectangular base.

EXAMPLE 5 Use Theorem 10.10 a. Find the value of each variable. SOLUTION PQRS is inscribed in a circle, so opposite angles are supplementary. a. m P + m R = 180o m Q + m S = 180o 75o + yo = 180o 80o + xo = 180o y = 105 x = 100

EXAMPLE 5 Use Theorem 10.10 b. Find the value of each variable. SOLUTION b. JKLM is inscribed in a circle, so opposite angles are supplementary. m J + m L = 180o m K + m M = 180o 2ao + 2ao = 180o 4bo + 2bo = 180o 4a = 180 6b = 180 b = 30 a = 45

GUIDED PRACTICE for Example 5 Find the value of each variable. 5. SOLUTION y = 112 x = 98

GUIDED PRACTICE for Example 5 Find the value of each variable. 6. SOLUTION c = 62 x = 10

Daily Homework Quiz Find the value of x . 1. ANSWER 38

Daily Homework Quiz Find the value of x . 2. ANSWER 56

Daily Homework Quiz Find the value of x . 3. ANSWER 44

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

Daily Homework Quiz Find the value of each variable. 5. ANSWER x = 5; y = 10