1. Angles Related to a Circle Section 10.5 Works Cited: By: Tara Mazurczyk “Geometry.” Glencoe. 19 May 2008. McDougal, Littell & Company. “Angles Related To a Circle.” Geometry for Enjoyment and Challenge. Evanston: n.p., 1991. 468-78. Roberts. “Formulas for Working with Angles in Circles.” http://regentsprep.org. 19 May 2008.

2. 1. Central Angle: A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle. Central Angle = Intercepted Arc <AOB is a central angle. Its intercepted arc is the minor arc from A to B. m<AOB = 80º

3. The measure of an inscribed angle or a tangent-chord angle is ½ the measure of its intercepted arc

4. A D B C is a central angle Example: A D B C Given: mAC = Find: Inscribed Angles An inscribed angle is an angle whose vertex is on a circle and whose sides are determined by two chords.

5. Tangent-chord Angles D F E DE is a tangent and EF is a chord is a tangent-chord angle A tangent-chord angle is an angle who vertex is on a circle and whose sides are determined by a tangent and a chord that intersect at the tangent’s point of contact Example: A C B Given: AB is tangent at B, Find:

6. Tangent-tangent Angles A B C AB and BC are tangents which means that Example: C D E Given: Find: A tangent-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by two tangents D

7. The measure of a chord-chord angle is ½ the sum of the measures of the arcs intercepted by the chord-chord angle and its vertical angle.

8. Chord-chord Angles A chord-chord angle is an angle formed by two chords that intersect inside a circle but not at the center. A B C D AC and BD are chords E Example: A B C D E Given: Find:

9. The measure of a secant-secant angle, a secant-tangent angle, or a tangent-tangent angle is ½ the difference of the measures of the intercepted arcs.

10. Secant-secant Angles A secant-secant angle is an angle whose vertex is outside a circle and whose sides are determined by two secants. A B C D E AC and CD are secants Example: A B C D E Given: Find:

11. Secant-tangent Angles A secant-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by a secant and a tangent. A B C D AB is a tangent and BD is a secant Example: Given: B C D A Find:

12. Practice Problems: 1.) A B C D E Given: Find: 2.) A B C D Given: Find: AB and BC are tangent to E E

13. 3.) Given: Find: A C D B 4.) C D A Find: Given: B

14. 5.) A BC D E F H G 1 2 3 4 Find: mAF mDE FG is tangent to H

15. 6.) A B C D G F E I H J K Given: L AB and AH are tangent to J Find: 1 2 3 4 5 6

16. Answers: 1.) 2.)

17. 3.)4.) 5.)6.)