Arcs and Inscribed Angles
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 shows examples of angles that are not inscribed angles.
Figure 2 Angles that are not inscribed angles.
Refer to Figure 3 and the example that accompanies it.
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, QS is a diameter. Find m ∠ R. m ∠ R = 90° (Theorem 72).
Figure 6 An inscribed angle which intercepts a semicircle.
Example 4: In Figure 7 of circle 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