Source - Khan Academy Inscribed and Central Angles

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 1An inscribed angle and its intercepted arc.


Figure 2 shows examples of angles that are not inscribed angles.







Figure 2Angles that are not inscribed angles.


∠ QRS is not an ∠ TWV is not an
inscribed angle,inscribed angle,
since its vertexsince its vertex
is not on the circle.is not on the circle.

Refer to Figure 3 and the example that accompanies it.







Figure 3A 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 4Finding the measure of an inscribed angle.







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







Figure 5Two inscribed angles with the same measure.







Example 3: In Figure 6 , QS is a diameter. Find m ∠ R. m ∠ R = 90° (Theorem 72).







Figure 6An inscribed angle which intercepts a semicircle.


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






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


Find each of the following.
  1. m ∠ CAD
  2. m
  3. m ∠ BOC
  4. m
  5. m ∠ ACB
  6. m ∠ ABC
CliffsNotes.com. Arcs and Inscribed Angles. 7 Feb 2013
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