1. Lesson 8-5: Angle Formulas
Circles
Angle Formulas
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 x° y° 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 FormulasQ 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