angles in a circle keystone geometry

Angles in a CircleKeystone Geometry

Types of AnglesThere 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

Intercepted ArcIntercepted 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.

Central Angle

Central Angle(of a circle)

Central Angle(of a circle)

NOT A Central Angle

(of a circle)

Definition: An angle whose vertex lies on the center of the circle.

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

Measuring a Central Angle

The measure of a central angle is equal to the measure of its intercepted arc.

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).

1 42 3

No! No!Yes! Yes!

Examples:

Measuring an Inscribed Angle

The measure of an inscribed angle is equal to half the measure of its intercepted arc.

CorollariesIf two inscribed angles intercept the same

arc, then the angles are congruent.

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

Corollary #3If a quadrilateral is inscribed in a circle, then

its opposite angles are supplementary.

** Note: All of the Inscribed Arcswill add up to 360

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.

Exterior AnglesAn 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 secantsA secant and a tangent

Two tangents

Exterior Angle TheoremThe measure of the angle formed is equal to ½ the difference of the intercepted arcs.

Exterior Angle Theorem

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.

Interior Angle Example