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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 .

In the circle, the two lines A C and A E intersect outside the circle at the point A .

m C A E = 1 2 ( m C E m B D )

Example:

In the circle shown, if m B D = 28 ° and m C E = 88 ° , then find m C A E .

Substitute the angle measures.

m C A E = 1 2 ( m C E m B D ) = 1 2 ( 88 ° 28 ° ) = 1 2 ( 60 ° ) = 30 °

Therefore, m C A E = 30 ° .