LECTURE8-5

Lesson 8-5:

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

Central Angle:

Lesson 8-5: 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:

Central Angle Theorem:

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

Slide 4:

Lesson 8-5: Angle Formulas 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°

Inscribed Angle:

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

Intercepted Arc:

Lesson 8-5: Angle Formulas 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.

Slide 7:

Lesson 8-5: Angle Formulas 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.

Slide 8:

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

Slide 9:

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

Interior Angle Theorem:

Lesson 8-5: Angle Formulas 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

Slide 11:

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

Exterior Angles:

Lesson 8-5: Angle Formulas 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

Exterior Angle Theorem:

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

Example: Exterior Angle Theorem:

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

Slide 15:

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

Slide 16:

Lesson 8-5: Angle Formulas 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.