Secants, Tangents, and Angle Measures 9-6 Secants, Tangents, and Angle Measures Objectives: To find the measures of angles formed by intersecting secants and tangents in relation to intercepted arcs..

Vocabulary Secant

Secant A line that intersects a circle in exactly two points is called a secant of the circle. A secant of circle contains a chord of the circle.

Theorem 9-11 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 its intercepted arc.

Theorem 9-12 If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Example 1 Find the value of x. Find the measure of angle AET.

Theorem 9-13 Case 1 – Two Secants 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.

Example 2

Theorem 9-13 Case 2 – A Secant and a Tangent 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.

Example 3

Theorem 9-13 Case 3 – Two Tangents 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.

Example 4