1. Classifying Angles with Circles
Case 1: Vertex is on the circle.
a b.

2. Classifying Angles with Circles
Case 2: Vertex is inside the circle.

3. Classifying Angles with Circles
Case 3: Vertex is outside the circle.
a b.

4. c.

5. Theorem
If a tangent and a secant (or a chord) intersect ON a circle at the point of tangency, then the measure of the angle formed is one half the measure of its intercepted arc.

6. The angle on the circle is half the measure of the intercepted arc.
75°

7. Theorem
The measure of an angle formed by two secants or chords that intersect in the interior of a circle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

8. It’s the average of the two intercepted arcs –
(80+40)/2 = 60°

9. Theorem
The measure of an angle formed by two secants intersecting in the exterior of a circle is one-half the difference of the measures of the intercepted arcs.

10. It’s half the difference: (80-20)/2 = 30°