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:
 The angle made where they intersect (angle APB above)
 The angle made by the
intercepted arc CD
 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
 In the figure above click on "reset" then "hide details"
 Drag the points C and D to new locations
 Calculate the angle P
 Click on "show details" to check your result
Other circle topics
General
Equations of a circle
Angles in a circle
Arcs
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