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Angle of Intersecting Secants Theorem

If two lines intersect outside a circle , then the measure of an angle formed by the two lines is one half the positive difference of the measures of the intercepted arcs .

Math diagram

In the circle, the two lines $\overset{\leftrightarrow}{A C}$ and $\overset{\leftrightarrow}{A E}$ intersect outside the circle at the point A .

$m ∠ C A E = \frac{1}{2} (m \overset{⌢}{C E} - m \overset{⌢}{B D})$

Example:

In the circle shown, if $m \overset{⌢}{B D} = 28 °$ and $m \overset{⌢}{C E} = 88 °$ , then find $m ∠ C A E$ .

Substitute the angle measures.

$\begin{array}{l} m ∠ C A E = \frac{1}{2} (m \overset{⌢}{C E} - m \overset{⌢}{B D}) \\ = \frac{1}{2} (88 ° - 28 °) \\ = \frac{1}{2} (60 °) \\ = 30 ° \end{array}$

Therefore, $m ∠ C A E = 30 °$ .