1. Chapter 10.5 Notes: Apply Other Angle Relationships in Circles
Goal: You will find the measures of angles inside or outside a circle.

2. How do you find the measure of an inscribed angle?
Theorem 10.11:
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.

3. Ex. 1: Line m is tangent to the circle
Ex.1: Line m is tangent to the circle. Find the measure of the red angle or arc.
Line m is tangent to the circle. Find x or y.
Ex.2: Ex.3:

4. Intersecting Lines and Circles
If two lines intersect a circle, they intersect in three places.
Theorem Angles Inside the Circle Theorem:
If two chords intersect inside a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

5. Ex.4: Find the value of x. Ex.5: Find the value of x.

6. Theorem 10.13 Angles Outside the Circle Theorem:
If a tangent and a secant, two tangents, or two secants intersect outside a circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.
Ex.6: Find the value of x.

7. Find the value of the variable.
Ex.7: Ex.8:
Ex.9: Ex.10: