It’s not too bad to find the measures of angles outside a circle which intercept the circles as secants or tangents.  Just remember this simple truth:

\(\theta  = \frac{1}{2}\left( {{\text{far arc}} - {\text{near arc}}} \right)\)

Hopefully you intuitively understand the difference between a far arc and a near arc, but just in case, let’s explain.  The far arc is the largest arc between the farthest two points of the rays of the angle that intercept the circle.  The near arc is the smallest arc between the closest two points of the rays of the angle that intercept the circle.  Visually, it will make sense, so let’s check out an example.

Example:   Find the measure of the arc or angle indicated.

SOLUTION:  Here the far arc is AP, and the near arc is BD.  Then

\(40^\circ  = \frac{1}{2}\left( {far arc - near arc} \right)\)

\(40^\circ  = \frac{1}{2}\left( {140^\circ  - x} \right)\)

Then

\(80^\circ  = 140^\circ  - x\)

so that

\(x = 60^\circ \)

It’s that simple.  Sometimes the far arc and near arc are a little more difficult to determine, as you will see in the next example, but the procedure is the same.

Example:  Find the measure of the arc or angle indicated.

SOLUTION:  Here, hopefully you can see that the far arc is KN, while the near arc is KM.  We have

\(70^\circ  = \frac{1}{2}\left( {far arc - near arc} \right)\)

\(70^\circ  = \frac{1}{2}\left( {x - 61^\circ } \right)\)

\(140^\circ  = x - 61^\circ \)

\(x = 201^\circ \)

Just remember to correctly determine the far arc and near arc.

Below you can download some free math worksheets and practice.