1. 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

2. INTERSECTIONS OUTSIDE OR INSIDE A CIRCLEB 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 outside a circle, the angles formed are related to the arcs they intercept.

3. 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:

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

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

6. 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

7. 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

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

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

10. 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.

11. 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

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

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

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