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Intersecting Secant Angles Theorem
The angle made by two secants intersecting outside a circle is half the difference between the intercepted arc measures.
Try this In the figure below, drag the orange dots to reposition the secants. Note how the angles are related. (Note: The angles are rounded off to whole numbers for clarity).

When two secants intersect outside a circle, there are three angle measures involved:

  1. The angle made where they intersect (angle APB above)
  2. The angle made by the intercepted arc CD
  3. The angle made by the intercepted arc AB
This theorem states that the angle APB is half the difference of the other two. Stated more formally: This is read as "The measure of the angle P is the measure of the arc CD minus the measure of the arc AB divided by 2"

Recall that the measure of an arc is the angle it makes at the center of the circle. To see this more clearly, click on "show central angles" in the diagram above. For more on this see Angle measure of an arc.

It works for tangents too

The theorem still holds if one or both secants is a tangent. In the figure above, drag point C to the right until it meets A. The top line is now a tangent to the circle, and points A and C are in the same location. But the theorem still holds using the measures of the arcs CD and AB in the same way as before.

Make both lines into tangents in this way, and convince yourself the theorem still works.

Things to try

  1. In the figure above click on "reset" then "hide details"
  2. Drag the points C and D to new locations
  3. Calculate the angle P
  4. Click on "show details" to check your result

Other circle topics

General

Equations of a circle

Angles in a circle

Arcs