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Mastering Inscribed Angle Theorems in Geometry

Learn to calculate angles inscribed in circles, central angles, intercepted arcs, corollaries, and theorems with detailed examples and assignments. Enhance your math skills today!

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Mastering Inscribed Angle Theorems in Geometry

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  1. 11-3Inscribed Angles Objective: To find the measure of an inscribed angle.

  2. Central Angle Definition: An angle whose vertex lies on the center of the circle. NOT A Central Angle (of a circle) Central Angle (of a circle) Central Angle (of a circle)

  3. Central Angle Theorem Y 110 110 O Z The measure of a center angle is equal to the measure of the intercepted arc. Center Angle Intercepted Arc Give is the diameter, find the value of x and y and z in the figure. Example:

  4. Vocabulary A Intercepted Arc Inscribed Angle C B

  5. Theorem 11-9 (Inscribed Angle Theorem) • The measure of an inscribed angle is half the measure of its intercepted arc. A B C

  6. Example 1: Using the Inscribed Angle Theorem P ao 60o T Q 30o S bo 60o R

  7. A ° 40 D ° 50 B ° y ° x C E Example 2: Find the value of x and y in the figure.

  8. Corollaries to the Inscribed Angle Theorem • Two inscribed angles that intercept the same arc are congruent. • An angle inscribed in a semicircle is a right angle. • The opposite angles of a quadrilateral inscribed in a circle are supplementary.

  9. An angle inscribed in a semicircle is a right angle. P 180 90 S R

  10. Example 3: Using Corollaries to Find Angle Theorem • Find the diagram at the right, find the measure of each numbered angle. 120o 1 60o 2 4 80o 3 100o

  11. Example 4: Find the value of x and y. 85 + x = 180 x = 95 80 + y = 180 y = 100 yo xo 85o 80o

  12. Theorem 11-10 • The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. B B D D C C

  13. Example 5: Using Theorem 11-10 • Find x and y. J 90o Q 35o xo yo L K

  14. Assignment Page 601 #1-24

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