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10-4 Inscribed Angles You found measures of interior angles of polygons. • Find measures of inscribed angles. • Find measures of angles of inscribed polygons. Find the measure of each arc • mDE • mAED • mEBD 54° E 180° 54° A D 306° C B What kind of angle is angle ECD? Central angle Definition His name was inscribed on the award. The square is inscribed in the circle. B An inscribed angle is an angle with its vertex on a circle and sides that contain chords of the circle. C A Intercepted Arc An intercepted arc has endpoints on the sides of an inscribed angle and lies in the interior of the inscribed angle R C Q Intercepted arc P Center P is on a side of the inscribed angle P Center P is inside the inscribed angle S P Center P is in the exterior of the inscribed angle Sizing Up Inscribed Angles A • Measure the central angle ACB • What is the measure of the arc AB? • Measure the inscribed angle ADB • Compare the measure of the central angle and the inscribed angle. C D B The measure of an inscribed angle is half the measure of its intercepted arc. A C B p. 723 p. 724 Find the measure of each angle or arc indicated by a variable. x° C 24° z° 160° 48° y° 80° 90° A. Find mX. Answer: mX = 43 B. = 2(52) or 104 A. Find mC. A. 47 B. 54 C. 94 D. 188 A D If two inscribed angles intercept the same arc, then they are congruent. B C p. 724 ALGEBRA Find mR. R S R and S both intercept . mR mS Definition of congruent angles 12x – 13 = 9x + 2 Substitution x =5 Simplify. Answer: So, mR = 12(5) – 13 or 47. ALGEBRA Find mI. A. 4 B. 25 C. 41 D. 49 An inscribed angle that intercepts a semicircle is a right angle. p. 725 ALGEBRA Find mB. ΔABC is a right triangle because C inscribes a semicircle. mA + mB + mC = 180 (x + 4) + (8x – 4) + 90 = 180 9x + 90 = 180 9x = 90 Angle Sum Theorem Substitution Simplify. Subtract 90 from each side. x = 10 Divide each side by 9. Answer: So, mB = 8(10) – 4 or 76. 10-4 Assignment Page 727, 11-18, 23-24

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