2 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 Angle2. On the Circle: Inscribed Angle3. In the Circle: Interior Angle4. Outside the Circle: Exterior Angle* The measure of each angle is determined by the Intercepted Arc
3 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.
4 Central AngleDefinition: 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 AngleInscribed 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:3124Yes!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 CorollariesIf two inscribed angles intercept the same arc, then the angles are congruent.
9 An angle inscribed in a semicircle is a right angle. Corollary #2An angle inscribed in a semicircle is a right angle.
10 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
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 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 tangentTwo tangents
13 Exterior Angle Theorem The measure of the angle formed is equal to ½ the difference of the intercepted arcs.