How to Determine the Measure of an Angle whose Vertex Is Outside a Circle
An angle that intersects a circle can have its vertex inside, on, or outside the circle. This article discusses the three types of angles that have their vertex outside a circle: secantsecant angles, secanttangent angles, and tangenttangent angles. A tangent is a line that touches a circle at a single point; a secant is a line that intersects a circle at two points.
So here are the three types of angles that have their vertex outside a circle:

Secantsecant angle: A secantsecant angle, like angle BDF in the above figure on the top left, is an angle whose vertex lies outside a circle and whose sides are two secants of the circle.

Secanttangent angle: A secanttangent angle, like angle GJK in the above figure on the top right, is an angle whose vertex lies outside a circle and whose sides are a secant and a tangent of the circle.

Tangenttangent angle: A tangenttangent angle, like angle LMN in the above figure on the bottom, is an angle whose vertex lies outside a circle and whose sides are two tangents of the circle.
Measure of an angle outside a circle: The measure of a secantsecant angle, a secanttangent angle, or a tangenttangent angle is onehalf the difference of the measures of the intercepted arcs. For example, in the above figure,
Note that you subtract the smaller arc from the larger (if you get a negative answer, you know you subtracted in the wrong order).
Now, using the above figure, solve this problem with the angleoutsideacircle formula:
Plug these expressions into the formula, and you’re home free: