A **secant** is a line that intersects a circle at two points. In the figure below, is formed by two secants. The angle intercepts two arcs and . In this post, we will prove that the measure of the angle formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by it.

To prove this theorem we will connect and use the Inscribed Angle Theorem and Exterior Angle Theorem.

**Angle Secant Theorem**

The angle measure formed by two secants intersecting outside a circle is half the difference of the arcs intercepted by it

**Proof**

Draw and let the measures of arcs and be and respectively. By the Inscribed Angle Theorem, the measure of the an angle inscribed in a circle is half the measure of its intercepted arc. Therefore, and .

Now, by the Exterior Angle Theorem,

.

Substituting, we have t.

Therefore,

which is what we want to show.