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Angles in a Circle Keystone Geometry 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 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. 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. Measuring a Central Angle The measure of a central angle is equal to the measure of its intercepted arc. 123° 70° 28° 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: 1 3 2 No! Yes! 4 No! Yes! Measuring an Inscribed Angle The measure of an inscribed angle is equal to half the measure of its intercepted arc. 82° Corollaries If two inscribed angles intercept the same arc, then the angles are congruent. Corollary #2 An angle inscribed in a semicircle is a right angle. Corollary #3 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary. 88 82° 98° 92° ** Note: All of the Inscribed Arcs will 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. C E D 106° B A 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 Exterior Angle Theorem The measure of the angle formed is equal to ½ the difference of the intercepted arcs. ° x -y Ð1 = 2 ° ° x -y Ð2 = 2 ° ° x -y Ð3 = 2 ° Exterior Angle Theorem In the given example: ÐACB is an exterior Green° - Re d ° angle, therefore ÐACB = . 2 265 ° - 95 ° ÐACB = 2 170° ÐACB = = 85° 2 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. BC + DE mÐ1 = 2 Interior Angle Example If BD = 40 ° and CE = 160 ° 40 + 160 200 then mÐ1= = = 100 ° 2 2 If BD = 28 ° and mÐ1 = 76 ° , find CE 28 + x then 76 = so 152 ° = 28 + x 2 so x=124 ° . °