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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.

Figure %: The inscribed angle measures half 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.

Figure %: An angle whose sides are a chord and a tangent segment
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.

Figure %: An angle whose vertex is in the interior of a circle
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.

Figure %: An angle whose vertex lies outside of a circle

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.

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