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We already know that in a circle the measure of a central
angle is equal to the measure of the
arc it intercepts. But what if the central angle
had its vertex elsewhere?

An angle whose vertex lies on a circle and whose sides intercept the circle (the
sides contain chords of the circle) is called an inscribed angle. The
measure of an inscribed angle is half the measure of the arc it intercepts.

If the vertex of an angle is on a circle, but one of the sides of the angle is
contained in a line tangent to the circle, the angle is no longer an inscribed
angle. The measure of such an angle, however, is equal to the measure of an
inscribed angle. It is equal to one-half the measure of the arc it intercepts.

The angle ABC is equal to half the measure of arc AB (the minor arc defined by
points A and B, of course).

An angle whose vertex lies in the interior of a circle, but not at its center,
has rays, or sides, that can be extended to form two secant
lines. These secant lines intersect each
other at the vertex of the angle. The measure of such an angle is half the sum
of the measures of the arcs it intercepts.

The measure of angle 1 is equal to half the sum of the measures of arcs AB and
DE.

When an angle's vertex lies outside of a circle, and its sides don't intersect
with the circle, we don't necessarily know anything about the angle. The
angle's sides, however, can intersect with the circle in three different ways.
Its sides can be contained in two secant lines, one secant line and one
tangent line, or two tangent lines. In any case,
the measure of the angle is one-half the difference between the measures of the
arcs it intercepts. Each case is pictured below.

In part (A) of the figure above, the measure of angle 1 is equal to one-half the
difference between the measures of arcs JK and LM. In part (B), the measure of
angle 2 is equal to one-half the difference between the measures of arcs QR and
SR. In part (C), the measure of angle 3 is equal to one-half the difference
between the measures of arcs BH and BJH. In this case, J is a point labeled
just to make it easier to understand that when an angle's sides are parts of
lines tangent to a circle, the arcs they intercept are the major and minor arc
defined by the points of tangency. Here, arc BJH is the major arc.