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 ∠ Rm ∠ R = 90° (Theorem 72).

Figure 6 An inscribed angle which intercepts a semicircle.

Example 4: In Figure 7 of circle Om   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

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