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Lesson 8-5: Angle Formulas

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1 Lesson 8-5: Angle Formulas

2 Lesson 8-5: Angle Formulas
Central Angle Definition: An angle whose vertex lies on the center of the circle. Central Angle (of a circle) Central Angle (of a circle) NOT A Central Angle (of a circle) Lesson 8-5: Angle Formulas

3 Lesson 8-5: Angle Formulas
Central Angle Theorem The measure of a center angle is equal to the measure of the intercepted arc. Y Z O 110 Intercepted Arc Center Angle 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 Lesson 8-5: Angle Formulas
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 No! Yes! No! Yes! Lesson 8-5: Angle Formulas

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

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

10 Interior Angle Theorem
Definition: Angles that are formed by two intersecting chords. 1 A B C D 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 Lesson 8-5: Angle Formulas
Example: Interior Angle Theorem 91 A C B D 85 Lesson 8-5: Angle Formulas

12 Lesson 8-5: Angle Formulas
Exterior Angles An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. 3 y x 2 1 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 Lesson 8-5: Angle Formulas
Q G F D E C 1 2 3 4 5 6 A 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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