You have learned how to find the measures of central angles and inscribed angles in circles. What if an angle is not inscribed and its not a central angle? There are three theorems that explain how to find these other types of angle measures. We love circles and angles!

I’ve got the first theorem!

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 its intercepted arc. 182° 91° 178° 89° What do you notice about the sum of the arc measures? What do you notice about the sum of the angle measures?

I’ve got the next theorem!

Theorem: If two secants intersect in the interior of a circle, then the measure of an angle formed is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. intersection  1 = ½ (intercepted arc + intercepted arc of vertical angle) 95° 105°  1 = ½ ( ) = ½ (200) = 100° 100° So, what is the measure of  3?  2?  4?

Let me do the third one!

Theorem: 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 ° 100° 160° 40° 130° 230°  1 = ½ ( )  1 = ½ (90)  1 = 45°  2 = ½ ( )  2 = ½ (120)  2 = 60°  3 = ½ ( )  3 = ½ (100)  3 = 50°

I hope that’s it! That’s a lot to remember! That slide way back at the beginning said 3 theorems - that was 3!

Colored note card Special segments and angle measures 1  1 = ½ (intercepted arc)  2 = ½ (intercepted arc + intercepted arc of vertical angle)  3 = ½ (positive difference of intercepted arcs)

Try a few. Find the measure of each angle. 84° 14° 170° 76° 290° 70° 15° 90° 260° 100° 80°

Try some more on your own! Can I, please?