Lesson 8 5 Angle Formulas Lesson 8 5

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

Central Angle Theorem The measure of a center angle is equal to the measure

Example: Find the measure of each arc. 4 x + 3 x + (3

Inscribed Angle: An angle whose vertex lies on a circle and whose sides are

Intercepted Arc: An angle intercepts an arc if and only if each of the

Inscribed Angle Theorem The measure of an inscribed angle is equal to ½ the

Examples: Find the value of x and y in the fig. F y A

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

Interior Angle Theorem Definition: Angles that are formed by two intersecting chords. A D

Example: Interior Angle Theorem A 91 C y° x° B D 85 Lesson 8

Exterior Angles An angle formed by two secants, two tangents, or a secant and

Exterior Angle Theorem The measure of the angle formed is equal to ½ the

Example: Exterior Angle Theorem Lesson 8 -5: Angle Formulas 14

Inscribed Quadrilaterals If a quadrilateral is inscribed in a circle, then the opposite angles

Slides: 16

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Lesson 8 -5 Angle Formulas Lesson 8 -5: Angle Formulas 1

Central Angle Definition: An angle whose vertex lies on the center of the circle. Central Angle (of a circle) Lesson 8 -5: Angle Formulas NOT A Central Angle (of a circle) 2

Central Angle Theorem The measure of a center angle is equal to the measure of the intercepted arc. Y Intercepted Arc Center Angle O 110 0 11 Example: Give is the diameter, find the value of x and y and z in the figure. Z Lesson 8 -5: Angle Formulas 3

Example: Find the measure of each arc. 4 x + 3 x + (3 x +10) + 2 x + (2 x-14) = 360° 14 x – 4 = 360° 14 x = 364° x = 26° 4 x = 4(26) = 104° 3 x = 3(26) = 78° 3 x +10 = 3(26) +10= 88° 2 x = 2(26) = 52° 2 x – 14 = 2(26) – 14 = 38° Lesson 8 -5: Angle Formulas 4

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 No! 3 2 Yes! No! Lesson 8 -5: Angle Formulas 4 Yes! 5

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. Lesson 8 -5: Angle Formulas 6

Inscribed Angle Theorem The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Y 0 11 Inscribed Angle 55 Z Intercepted Arc An angle formed by a chord and a tangent can be considered an inscribed angle. Lesson 8 -5: Angle Formulas 7

Examples: Find the value of x and y in the fig. F y A A 40 D B B 50 y x x C C E E Lesson 8 -5: Angle Formulas 8

An angle inscribed in a semicircle is a right angle. P S 180 90 R Lesson 8 -5: Angle Formulas 9

Interior Angle Theorem Definition: Angles that are formed by two intersecting chords. A D 1 Interior Angle Theorem: E 2 B C The measure of the angle formed by the two intersecting chords is equal to ½ the sum of the measures of the intercepted arcs. Lesson 8 -5: Angle Formulas 10

Example: Interior Angle Theorem A 91 C y° x° B D 85 Lesson 8 -5: Angle Formulas 11

Exterior Angles An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle. x y 1 x y 2 x Two secants y 3 2 tangents A secant and a tangent Lesson 8 -5: Angle Formulas 12

Exterior Angle Theorem The measure of the angle formed is equal to ½ the difference of the intercepted arcs. Lesson 8 -5: Angle Formulas 13

Example: Exterior Angle Theorem Lesson 8 -5: Angle Formulas 14

D C 6 30° E A 3 100° Q 2 1 5 4 25° F G Lesson 8 -5: Angle Formulas 15

Inscribed Quadrilaterals If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary. m DAB + m DCB = 180 m ADC + m ABC = 180 Lesson 8 -5: Angle Formulas 16