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.

angle secant theorem

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}.

angle secant theorem 2

Now, by the Exterior Angle Theorem,

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

Substituting, we have tm \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.

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