6.5Apply 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

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

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

6.5Apply 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

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

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

6.5Apply 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

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

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

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