1. Inscribed Angles Find measures of inscribed angles Find measures of angles of inscribed polygons. Three congruent central angles are pictured. What is the measure of each angle?

2. INSCRIBED ANGLES A B C D An inscribed angle is an angle that has its vertex on the circle and its sides contained in chords of the circle. Vertex B is on the circle AB and BC are chords of the circle ADC is the arc intercepted by  ABC

3. Theorem Inscribed Angle Theorem If an angle is inscribed by a circle then the measure of the angle equals one-half the measure of the intercepted arc (or the measure of the intercepted arc is twice the measure of the inscribed angle). A B C D

4. Example 1 Measures of Inscribed Angles A B C D O 1 2 3 4 5 In circle O, measure of arc AB = 140, measure of arc BC = 100, and measure of arc AD equals the measure of arc DC. Find the measures of the numbered angles. A B C D O 1 2 34 5

5. Example 1 Measures of Inscribed Angles A B C D O 1 2 3 4 5 In circle O, measure of arc AB = 140, measure of arc BC = 100, and measure of arc AD equals the measure of arc DC. Find the measures of the numbered angles. First find mDC and mAD

6. Theorem If two inscribed angles of a circle (or congruent circles) intercept congruent arcs or the same arc, then the angles are congruent. AB CD A F E B C D

7. Example 2 Inscribed Arcs and Probability

8. ANGLES OF INSCRIBED POLYGONS An inscribed triangle with a side that is a diameter is a special type of triangle.

9. Theorem If an inscribed angle intercepts a semicircle, the angle is a right angle. A B C D

10. Example 3 Angles of an Inscribed Triangle A B F D E Triangles ABD and ADE are inscribed in circle F with AB  BD. Find the measure of each numbered angle if m  1 = 12x – 8 and m  2 = 3x + 8. 1 2 3 4

11. Example 3 Angles of an Inscribed Triangle A B F D E Triangles ABD and ADE are inscribed in circle F with AB  BD. Find the measure of each numbered angle if m  1 = 12x – 8 and m  2 = 3x + 8. 12x – 8 64° 3x + 8 3 4 26°