1 1/3/13 Unit 4 Polygons and Circles Angle Formulas

2 Central Angle (of a circle) Central Angle (of a circle) NOT A Central Angle (of a circle) Central Angle An angle whose vertex lies on the center of the circle. Definition:

3 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.

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°

5 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) No! Yes! Examples:

6 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.

7 Inscribed Angle Theorem The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Y Z 55  110  Inscribed Angle Intercepted Arc An angle formed by a chord and a tangent can be considered an inscribed angle.

8 Examples: Find the value of x and y in the fig. y  40  x  50  A B C D E y  x   A B C E F

9 An angle inscribed in a semicircle is a right angle. R P 180  S 90 

10 Interior Angle Theorem Angles that are formed by two intersecting chords.Definition: The measure of the angle formed by the two intersecting chords is equal to ½ the sum of the measures of the intercepted arcs. Interior Angle Theorem: 1 A B C D E 2

11 A B C D x°x° 91  85  Example: Interior Angle Theorem y°y°

12 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 y  x  1 x  y  Two secants A secant and a tangent 2 tangents

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

14 Example: Exterior Angle Theorem

15 Q G F D E C A 100° 30° 25°

16 Inscribed Quadrilaterals m  DAB + m  DCB = 180  m  ADC + m  ABC = 180  If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary.