# The Angle Secant Theorem

A secant is a line that intersects a circle at two points. In the figure below, $\angle E$ is formed by two secants. The angle intercepts two arcs $\overline{AB}$ and $\overline{CD}$. 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 $\overline{BC}$ 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 $\overline{BC}$ and let the measures of arcs $AB$ and $CD$ be $\alpha$ and $\beta$ 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, $\angle C = \frac{\alpha}{2}$ and $\angle CBD = \frac{\beta}{2}$.

Now, by the Exterior Angle Theorem,

$m \angle E + m \angle C = m \angle CBD$.

Substituting, we have t$m \angle E + \displaystyle \frac{\alpha}{2} = \displaystyle \frac{\beta}{2}$.

Therefore, $m \angle E = \displaystyle \frac{\beta}{2} - \displaystyle \frac{\alpha}{2}$

which is what we want to show.