Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas 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) Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas 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. Lesson 8-5: Angle Formulas

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° Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas 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: 3 1 2 4 No! Yes! No! Yes! Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas 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. Lesson 8-5: Angle Formulas

Lesson 8-5: Angle Formulas Inscribed Angle Theorem The measure of an inscribed angle is equal to ½ the measure of the intercepted arc. Y Inscribed Angle 110 55 Z Intercepted Arc An angle formed by a chord and a tangent can be considered an inscribed angle. Lesson 8-5: Angle Formulas

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

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

Interior Angle Theorem Definition: Angles that are formed by two intersecting chords. 1 A B C D 2 E Interior Angle Theorem: 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

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

Lesson 8-5: Angle Formulas 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 1 Two secants 2 tangents A secant and a tangent Lesson 8-5: Angle Formulas

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

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

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

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