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Understanding Angle Formulas in Circles: Lessons & Examples

Learn about central angles, intercepted arcs, inscribed angles, and more in this lesson on circle angle formulas. Explore how to find angle measures and apply theorems to solve problems. Enhance your geometry skills today!

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Understanding Angle Formulas in Circles: Lessons & Examples

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  1. Learning Target 8 Angle Formulas Lesson 8-5: Angle Formulas

  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) Lesson 8-5: Angle Formulas

  3. Y 110 110 O Z Central Angle Theorem The measure of a center angle is equal to the measure of the intercepted arc. Center Angle Intercepted Arc Example: Give is the diameter, find the value of x and y and z in the figure. Lesson 8-5: Angle Formulas

  4. Example: Find the measure of each arc. 4x + 3x + (3x +10) + 2x + (2x-14) = 360° 14x – 4 = 360° 14x = 364° x = 26° 4x = 4(26) = 104° 3x = 3(26) = 78° 3x +10 = 3(26) +10= 88° 2x = 2(26) = 52° 2x – 14 = 2(26) – 14 = 38° Lesson 8-5: Angle Formulas

  5. Inscribed Angle Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle). Examples: 3 1 2 4 Yes! No! No! Yes! Lesson 8-5: Angle Formulas

  6. Intercepted Arc Intercepted Arc: An angle intercepts an arc if and only if each of the following conditions holds: 1. The endpoints of the arc lie on the angle. 2. All points of the arc, except the endpoints, are in the interior of the angle. 3. Each side of the angle contains an endpoint of the arc. Lesson 8-5: Angle Formulas

  7. Inscribed Angle Theorem The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Y Inscribed Angle 110 55 Z Intercepted Arc An angle formed by a chord and a tangent can be considered an inscribed angle. Lesson 8-5: Angle Formulas

  8. A F A ° ° 40 y D ° 50 B ° ° y 50 B ° x C ° C x E E Examples: Find the value of x and y in the fig. Lesson 8-5: Angle Formulas

  9. An angle inscribed in a semicircle is a right angle. P 180 90 S R Lesson 8-5: Angle Formulas

  10. A D 1 B C Interior Angle Theorem Definition: Angles that are formed by two intersecting chords. 2 E Interior Angle Theorem: The measure of the angle formed by the two intersecting chords is equal to ½ the sum of the measures of the intercepted arcs. Lesson 8-5: Angle Formulas

  11. Example: Interior Angle Theorem 91 A C x° y° B D 85 Lesson 8-5: Angle Formulas

  12. ° 1 y ° 2 y ° ° x 3 x ° y ° x Exterior Angles An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. Two secants 2 tangents A secant and a tangent Lesson 8-5: Angle Formulas

  13. Exterior Angle Theorem The measure of the angle formed is equal to ½ the difference of the intercepted arcs. Lesson 8-5: Angle Formulas

  14. Example: Exterior Angle Theorem Lesson 8-5: Angle Formulas

  15. D 6 C E Q 5 3 A F 2 1 4 G 30° 25° 100° Lesson 8-5: Angle Formulas

  16. Inscribed Quadrilaterals If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. mDAB + mDCB = 180  mADC + mABC = 180  Lesson 8-5: Angle Formulas

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