10.4 Other Angle Relationships in Circles

1. 10.4 Other Angle Relationships in Circles

2. Theorems • If a tangent and a chord intersect at a point ON the circle, then the measure of each angle formed is ½ the measure of its intercepted arc m<1 = ½ mRS m< 2 = ½ mSHR

3. Examples of Theorem 1 • Line m is tangent to the circle. Find m<1 • M<1 = 39 • Line m is tangent to the circle. Find arc DA • 2) 10x = 9x + 20 • x = 20 • Arc DA = 200

4. Theorem 2 2. If two chords intersect in the interior of a circle, then the measure of each angle is ½ the sum of the measures of the arcs intercepted by the angle and its vertical angle.

5. Example of Theorem 2 • Find x • X = ½ (174 + 106) • X = 1/2 (280) • X = 140

6. Theorem 3 (Last one!) • If a tangent and a secant, two tangents, or 2 secants intersect in the exterior of a circle, then the measure of the angle formed is ½ the difference of the measures of the intercepted arcs m<1 = ½ (mOC – mDC) m<2 = ½ (mWHO – mWO) m<3 = ½ (mSR – mTO)

7. Example of Theorem 3 • Find x • From last page: m<1 = ½ (mOC – mDC) • For this problem: • 72 = ½ (200 – x) • 72 = 100 – ½ x • -28 = -1/2 x • X = 56

8. Assignment p. 624 2-34even