1. Apply Other Angle Relationships
6.5
Apply Other Angle Relationships
Theorem 6.13
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. A C B

2. Apply Other Angle Relationships
6.5
Apply Other Angle Relationships
Example 1
Find angle and arc measures
Line m is tangent to the circle. Find the indicated measure. m A B m F D E
Solution

3. Apply Other Angle Relationships
6.5
Apply Other Angle Relationships
Checkpoint. Find the indicated measure. A B C

4. Apply Other Angle Relationships
6.5
Apply Other Angle Relationships
Theorem 6.14
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. D C A B

5. Apply Other Angle Relationships
6.5
Apply Other Angle Relationships
Example 2
Find an angle measure inside a circle J H F G
Find the value of x.
Solution
The chords FH and GJ intersect inside the circle.

6. Apply Other Angle Relationships
6.5
Apply Other Angle Relationships
Checkpoint. Find the value of x. J H F G

7. Apply Other Angle Relationships
6.5
Apply Other Angle Relationships
Theorem 6.15
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. B A C Q P R X Y W Z

8. Apply Other Angle Relationships
6.5
Apply Other Angle Relationships
Example 3
Find an angle measure outside a circle J F H G
Find the value of x.
Solution
The tangent GF and the secant GJ intersect outside the circle.

9. Apply Other Angle Relationships
6.5
Apply Other Angle Relationships
Checkpoint. Find the value of y. J K H G F

10. Apply Other Angle Relationships
6.5
Apply Other Angle Relationships
Pg. 225, 6.5 #1-23