10 6 Secants Tangents Angle Measures Objectives Find

10. 6 Secants, Tangents, & Angle Measures

Objectives Find measures of angles formed by lines intersecting on, inside, or outside a circle.

Secants and Interior Angles Theorem 10. 12: If two secants intersect in the interior of a , then the measure of an formed is ½ the measure of the sum of the arcs intercepted by the secants that created the . A C m AOC = ½ (m arc AC + m arc DB) m AOD = ½ (m arc AD + m arc CB) D O B

Example 1: Find Method 1 if and

Example 1: Method 2 Answer: 98

Your Turn: Find if Answer: 138 and

Secants, Tangents and Angles Theorem 10. 13: If a secant and a tangent intersect at the point of tangency, then the measure of each formed is ½ the measure of its intercepted arc. D A O C m DOB = ½ (m arc AB) m COB = ½ (m arc CFB) B F

Example 2: Find Answer: 55 if and

Your Turn: Find Answer: 58 if and

Secants – Tangents & Exterior Angles Theorem 10. 14: If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the formed is ½ the measures of the difference of the intercepted arcs. A G E m BOC = ½ (m arc BC – m arc HI) m AOC= ½ (m arc AC – m arc AI) m AED= ½ (m arc ABD – m arc AD) O F I B H D C

Example 3: Find x. Theorem 10. 14 Multiply each side by 2. Add x to each side. Subtract 124 from each side. Answer: 17

Your Turn: Find x. Answer: 111

Example 4: A jeweler wants to craft a pendant with the shape shown. Use the figure to determine the measure of the arc at the bottom of the pendant. Let x represent the measure of the arc at the bottom of the pendant. Then the arc at the top of the circle will be 360 – x. The measure of the angle marked 40° is equal to one-half the difference of the measure of the two intercepted arcs.