Warm-Up A circle and an angle are drawn in the same plane. Find all possible ways in which the circle and angle intersect at two points.

9.4 – Angles Formed by Secants and Tangents Topics: Theorems about measures of arcs intercepted by angles

Classification of Angles with Circles Case 1: Vertex is on the circle Two secants Secant and tangent

Classification of Angles with Circles Case 2: Vertex is inside the circle Two secants

Classification of Angles with Circles Case 3: Vertex is outside the circle Two secants Secant and tangent Two tangents

Vertex on Circle – Secant and Tangent

Vertex on Circle – Secant and Tangent Part 1 – The secant-tangent angle is a right angle. P V C

Vertex on Circle – Secant and Tangent Part 2 – The secant-tangent angle is acute. A P 3 1 Copy and complete the table. 2 V C

Vertex on Circle – Secant and Tangent Part 3 – The secant-tangent angle is obtuse. P 1 3 A Copy and complete the table. 2 V C

Theorem If a tangent and a secant (or a chord) intersect on a circle at the point of tangency, then the measure of the angle formed is _______ the measure of its intercepted arc. one-half

Vertex Inside Circle – Two Secants Find the relationship between and the measures of A D 2 1 V P B Copy and complete the table. C

Theorem The measure of an angle formed by two secants or chords that intersect in the interior of a circle is _______ the _______ of the measures of the arcs intercepted by the angle and its vertical angle. one-half sum

Vertex Outside Circle – Two Secants Find the relationship between and the measures of B 2 A 1 V C Copy and complete the table. D

Theorem The measure of an angle formed by two secants that intersect in the exterior of a circle is ________ the _________ of the measures of the intercepted arcs. one-half difference

Example 1 Given: AB is tangent to circle P at A, mAC = 60, mCD = 70, and mDE = 80 Find mDAB, mCBA, and mDGC. Because the vertex of DAB is ON the circle, . mDAB

Example 1 Given: AB is tangent to circle P at A, mAC = 60, mCD = 70, and mDE = 80 Find mDAB, mCBA, and mDGC. The vertex of CBA is outside the circle, so mCBA

Example 1 Given: AB is tangent to circle P at A, mAC = 60, mCD = 70, and mDE = 80 Find mDAB, mCBA, and mDGC. The vertex of DGC is inside the circle, so mDGC

Practice Problem Given: XY is tangent to circle T at X, mUV = 90, mVX = 130, and mUW = 20 V X T W Z Y 1 2 Find mWXY, m1, and m2.

Homework p.593 #10-26