Vertex at the Center of a Circle
The measure of the angle is equal to the measure of the arc

Vertex on the Circle
The measure of the angle whose vertices are on the circle is half the measure of the interecpted arc.

Definition- An inscribed angle is an angle whose vertex is on a circle and whose sides are determined by two chords

Definition- A tangent-chord angle is an angle whose vertex is on a circle and whose sides are determined by a tangent and a chord that intersect at the point of contact.

Vertex inside But Not at the Center of a Circle

Definition- A chord-chord angle is formed by two chords that intersect inside a circle, but not at the center

Theorem- The measure of a chord-chord angle equals half of the sum of the measures of the arcs cut off by the chord-chord angle and its vertical angle.

Vertex Outside of a Circle
The measure of an angle whose vertex is outside a circle is half of the difference of the two intercepted arcs.

Definition- A secant-secant angle whose vertex is outside a circle and whose sides are determined by two secants.

Definition- A secant-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by a secant and a tangent.

Definition- A tangent-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by two tangents.