1. Angles in a Circle Keystone Geometry

2. Types of Angles
There are four different types of angles in any given circle. The type of angle is determined by the location of the angles vertex.
1. In the Center of the Circle: Central Angle
2. On the Circle: Inscribed Angle
3. In the Circle: Interior Angle
4. Outside the Circle: Exterior Angle
* The measure of each angle is determined by the Intercepted Arc

3. 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.

4. 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)
* The measure of a central angle is equal to the measure of the intercepted arc.

5. Measuring a Central Angle
The measure of a central angle is equal to the measure of its intercepted arc.

6. 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!
Yes!
No!

7. Measuring an Inscribed Angle
The measure of an inscribed angle is equal to half the measure of its intercepted arc.

8. Corollaries
If two inscribed angles intercept the same arc, then the angles are congruent.

9. An angle inscribed in a semicircle is a right angle.
Corollary #2
An angle inscribed in a semicircle is a right angle.

10. Corollary #3
If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.
** Note: All of the Inscribed Arcs
will add up to 360

11. Another Inscribed Angle
The measure of an angle formed by a chord and a tangent is equal to half the measure of the intercepted arc.

12. Exterior Angles
An exterior angle is formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. The vertex lies outside of the circle.
Two secants
A secant and a tangent
Two tangents

13. Exterior Angle Theorem
The measure of the angle formed is equal to ½ the difference of the intercepted arcs.

14. Exterior Angle Theorem

15. Interior Angles
An interior angle can be formed by two chords (or two secants) that intersect inside of the circle.
The measure of the angle formed is equal to ½ the sum of the intercepted arcs.

16. Interior Angle Example