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Warm-Up Exercises Bell Work 1. 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. 1 3. Solve 3x = ( 4x + 12). 2 Warm-Up Exercises Notes Definitions An inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc of the angle. Warm-Up Exercises Notes Theorem 10.7 Measure of an Inscribed Angle Theorem The measure of an inscribed angle is one half the measure of its intercepted arc. Warm-Up1Exercises EXAMPLE Find the indicated measure in a. m T b. mQR P. Warm-Up2Exercises EXAMPLE Find mRS and m and RUS? STR. What do you notice about STR Warm-Up Exercises Notes Theorem 10.8 If two inscribed angles of a circle intercept the same arc, then the angles are congruent. Warm-Up Exercises GUIDED PRACTICE Find the measure of the red arc or angle. 1. Warm-Up Exercises GUIDED PRACTICE Find the measure of the red arc or angle. 2. Warm-Up Exercises GUIDED PRACTICE Find the measure of the red arc or angle. 3. Warm-Up Exercises Notes Definitions A polygon is an inscribed polygon if all of its vertices lie on the circle. The circle that contains the vertices is a circumscribed circle. Warm-Up Exercises Notes Theorem 10.9 If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the right angle. Warm-Up4Exercises EXAMPLE 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? Warm-Up4Exercises EXAMPLE 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. Warm-Up Exercises 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. SOLUTION Make the diameter of your circle the diagonal of the rectangular base. Warm-Up Exercises Notes Theorem 10.10 A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. (not all quadrilaterals can be inscribed in a circle) Warm-Up5Exercises EXAMPLE Find the value of each variable. a. Warm-Up Exercises GUIDED PRACTICE Find the value of each variable. 6. Daily Homework Warm-Up Exercises Quick CheckQuiz Find the value of x . 1. Daily Homework Warm-Up Exercises Quick CheckQuiz Find the value of x . 2. Daily Homework Warm-Up Exercises Quick CheckQuiz Find the value of x . 3. Daily Homework Warm-Up Exercises Quick CheckQuiz Find the value of each variable. 4. Daily Homework Warm-Up Exercises Quick CheckQuiz Find the value of each variable. 5. Warm-Up Exercises For Tomorrow Homework HW: 10.4 (pg. 666-667) #2,8,12,14,17,20-25