Secants, Tangents and Angle Measures Find measures of angles formed by lines intersecting on or inside a circle. Find measures of angles formed by lines intersecting outside the circle. Water droplet on a CD

INTERSECTIONS ON OR INSIDE A CIRCLE B F E D A line that intersects a circle in exactly two points is called a secant. In the figure above, SF and EF are secants of the circle. When two secants intersect inside a circle, the angles formed are related to the arcs they intercept.

Theorem If two secants intersect in the interior of a circle, then the measure of the angle formed is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle. A D 2 1 B C Examples:

Example 1 Secant-Secant Angle Find m2 if mBC = 30 and mAD = 20 B 2 30° A 1 20° C D E

Example 2 Secant-Secant Angle F Find m4 if mFG = 88 and mEH = 76 88° G 4 3 E H 76°

A secant can also intercept a tangent at the point of tangency. Each angle formed has a measure half that of the arc it intercepts. B A D C E

Theorem If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one half the measure of the intercepted arc. B A D C E

Example 3 Secant-Tangent Angle Find the mABC if mAB = 102° 102°

Example 4 Secant-Tangent Angle Find the mRPS if mPT = 114° and mTS = 136° 114° 136°

INTERSECTIONS OUTSIDE A CIRCLE Secants and tangents can also intersect outside a circle. The measure of the angle formed also involves half the measure of the arcs they intercept.

Theorem D D C E B C C B B D A A A If two secants, a secant and a tangent, or two tangents intersect in the exterior of a circle, then the measure of the angle formed is one-half the positive difference of the measures of the intercepted arcs. Two Secants Secant-Tangent Two Tangents D D C E B C B C B D A A A

Example 5 Secant-Secant Angle Find x x° 120° 50°

Example 6 Secant-Secant Angle Find x 62° 141° x°

Example 7 Secant-Secant Angle Find x 11° x°