advertisement

CENTRAL AND INSCRIBED ANGLES #18 A central angle is an angle whose vertex is the center of a circle and whose sides intersect the circle. The degree measure of a central angle is equal to the degree measure of its intercepted arc. For the circle at right with center C, ∠ACB is a central angle. An INSCRIBED ANGLE is an angle with its vertex on the circle and whose sides intersect the circle. The arc formed by the intersection of the two sides of the angle and the circle is called an ! is an INTERCEPTED ARC. ∠ADB is an inscribed angle, AB intercepted arc. The INSCRIBED ANGLE THEOREM says that the measure of any inscribed angle is half the measure of its intercepted arc. Likewise, any intercepted arc is twice the measure of any inscribed angle whose sides pass through the endpoints of the arc. m∠ADB = 1 2 A B C A • • B • D ! = 2m∠ADB ! and AB AB Example 1 Example 2 ! and P, m !CPQ = 70˚ . Find mCQ !. mCRQ In the circle shown below, the vertex of the angles is at the center. Find x. In C B 2x R P Q C ! = m !CPQ = 70˚ mCQ ! = 360° ! mCQ " = 360° = 70° = 290° mCRQ A x Since AB is the diameter of the circle, x + 2x = 180˚ ⇒ 3x = 180˚ ⇒ x = 60˚ Example 3 Solve for x. ˚ 100 A B x C ! = 220˚, mBC ! = 360˚ – 220˚ = 140˚. The Since mBAC ! or 70˚. m!BAC equals half the mBC 120˚ GEOMETRY Connections © 2007 CPM Educational Program. All rights reserved. 51 P if m!WPX = 28˚ , m!ZPY = 38˚ , and WZ and XV are Find each measure in diameters. ! 1. YZ 5. !XPY ! 2. WX 6. ! XY 3. !VPZ 7. ! XWY ! 4. VWX 8. ! WZX W P V X Z Y In each of the following figures, O is the center of the circle. Calculate the values of x and justify your answer. 9. 10. O 11. 136° O x 146° O 49° x 13. 14. 12. O O • O 18° x x 16. 150° x O 62° 15. 100° O x • x x 27° 100° 17. 18. 55° O • 19. 12° x O 20. O • • 35° x 91° x x O • 29° 101° Answers 1. 38˚ 2. 28˚ 3. 28˚ 4. 180˚ 5. 114˚ 6. 114˚ 7. 246˚ 8. 332˚ 9. 68˚ 10. 73˚ 11. 98˚ 12. 124˚ 13. 50˚ 14. 55˚ 15. 18˚ 16. 27˚ 17. 55˚ 18. 77˚ 19. 35˚ 20. 50˚ 52 © 2007 CPM Educational Program. All rights reserved. Extra Practice