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Inscribed Angles

B. A. O. C. AB is the intercepted arc of C. Inscribed Angles. The vertex of C is on circle O. The sides of C are chords of circle O. C is an inscribed angle. E. D. O. F. Polygons and Circles. A polygon is inscribed in a circle if all its vertices lie on the circle.

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Inscribed Angles

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  1. B A O C AB is the intercepted arc of C . Inscribed Angles The vertex of C is on circle O. The sides of C are chords of circle O. C is an inscribed angle.

  2. E D O F Polygons and Circles A polygon is inscribed in a circle if all its vertices lie on the circle. Circle O is circumscribed aboutDEF.

  3. A B BCD Which angle intercepts DAB? C D O Example 1 Which arc does A intercept? C Is quadrilateral ABCD inscribed in the circle? Yes Which angles appear to intercept major arcs? B and C What kind of angles do B and C appear to be? obtuse

  4. A B C mB = ½ mAC Inscribed Angle Theorem The measure of an inscribed angle is half the measure of its intercepted arc:

  5. a° P mPRS = ½ mPS mPQT = ½ mPT T 30° 60° S Q b° mPRS = ½ mPT + mTS R Example 2 Find the values of a and b: 60 = ½ a a = 120 b = ½ (120 + 30) b = ½ (150) = 75

  6. a° P mPRS = ½ mPS mPQT = ½ mPT T 25° 70° S Q b° mPRS = ½ mPT + mTS R Example 3 Find the values of a and b: 70 = ½ a a = 140 b = ½ (140 + 25) b = ½ (165) = 82.5

  7. A D C B Corollaries to the Inscribed Angle Theorem Corollary 1: Two inscribed angles that intercept the same arc are congruent. C  D

  8. O O Corollaries to the Inscribed Angle Theorem Corollary 2: An angle inscribed in a semicircle is a right angle.

  9. A D O C B Corollaries to the Inscribed Angle Theorem Corollary 3: The opposite angles of a quadrilateral inscribed in a circle are supplementary. A and C are supplementary. B and D are supplementary.

  10. A B F O E C D Example 4 Name a pair of congruent inscribed angles: FAD and FBD Name a right angle: FBC Name a pair of supplementary inscribed angles: FED and FBD

  11. v° 96° w° x° 72° z° y° Example 5 Find the values of of the variables: v = 180 – 96 = 84 w = ½ (84) = 42 x = ½ (72) = 36 y = 180 – 72 = 108 z = ½ (84) = 42

  12. z° 107° 98° x° w° y° Example 6 Find the values of of the variables: w = ½ (107) = 53.5 x = ½ (180 – 98) = ½ (82) = 41 y = 180 - 107 = 73 z = ½ (98 + 73) = ½ (171) = 85.5

  13. B D C B mC = ½ mBDC D C Angles Formed by Tangents and Chords The measure of an angle formed by a chord and a tangent that intersect on a circle is half the measure of the intercepted arc.

  14. A y° x° 58° C B mCB = 2(32) = 64 z° D Example 7 CD is tangent to circle O at C. AB is a diameter. Find the values of the variables. x = 90 (the angle inscribed in a semicircle is a right angle) y = 90 - 58 = 32 z = ½ (64) = 32

  15. J Q 35° x° z° y° z = ½ mJL = mQ = 35 L K Example 8 JK is tangent to circle O at J. QL is a diameter. Find the values of the variables. x = 90 y = 90 - 35 = 55

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