Central angles are probably the angles most often associated with a circle, but by no means are they the only ones. Angles may be inscribed in the circumference of the circle or formed by intersecting chords and other lines.

**Inscribed angle:**In a circle, this is an angle formed by two chords with the vertex on the circle.

**Intercepted arc:**Corresponding to an angle, this is the portion of the circle that lies in the interior of the angle together with the endpoints of the arc.

In Figure 1*ABC* is an inscribed angle and is its intercepted arc.

**Figure 1 **An inscribed angle and its intercepted arc.

Figure 2 *not* inscribed angles.

**Figure 2 **Angles that are not inscribed angles.

Refer to Figure 3

**Figure 3 **A circle with two diameters and a (nondiameter) chord.

Notice that *m* ∠3 is exactly half of *m* , and *m* ∠4 is half of *m* ∠3 and ∠4 are inscribed angles, and and are their intercepted arcs, which leads to the following theorem.

*Theorem 70:* The measure of an inscribed angle in a circle equals half the measure of its intercepted arc.

The following two theorems directly follow from *Theorem 70*.

*Theorem 71:* If two inscribed angles of a circle intercept the same arc or arcs of equal measure, then the inscribed angles have equal measure.

*Theorem 72:* If an inscribed angle intercepts a semicircle, then its measure is 90°.

**Example 1:** Find *m* ∠ *C* in Figure 4

**Figure 4 **Finding the measure of an inscribed angle.

**Example 2:** Find *m* ∠ *A* and *m* ∠ *B* in Figure 5

**Figure 5 **Two inscribed angles with the same measure.

**Example 3:** In Figure 6*m* ∠ *R*. *m* ∠ *R* = 90° *(Theorem 72).*

**Figure 6 ***An inscribed angle which intercepts a semicircle.*

**Example 4:** In Figure 7 *O*, *m* 60° and *m* ∠1 = 25°.

**Figure 7 **A circle with inscribed angles, central angles, and associated arcs.

Find each of the following.

a. *m* ∠ *CAD*

*b*.* m*

c.* m* ∠ *BOC*

d*. m*

e*. m* ∠ *ACB*

f.* m* ∠ *ABC*