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Inscribed Angles and Polygons: Understanding Measurements and Theorems

Learn about inscribed angles in a circle and their intercepted arcs, as well as the measurements and theorems related to them. Explore examples and practice problems.

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Inscribed Angles and Polygons: Understanding Measurements and Theorems

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  1. Do Now

  2. Geometry 10.4: Use Inscribed Angles and Polygons

  3. Inscribed Angle • 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. BCA is an inscribed angle. BA is an intercepted arc of BCA. B W C A

  4. Measure of an Inscribed Angle • The measure of an inscribed angle is half the measure of its intercepted arc. If m BCA = 20o Then m BA = 40o B W C A

  5. Theorem 10.8 • If two inscribed angles of a circle intercept the same arc, then the angles are congruent. B If m BA = 40o then m BCA = 20o and m BDA = 20o W C A D

  6. Theorem 10.9 • If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle. (The converse is also true.) A B W C

  7. Examples

  8. Theorem 10.10 • A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary. Angles B and D are supplementary And Angles A and C are supplementary B A W D C

  9. Examples

  10. Do Now: 10.4 Review

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