9 6 Secants Tangents and Angle Measures Objectives

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 onehalf 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