Arcs And Inscribed Angles

There are several different angles associated with circles. Perhaps the one that most immediately comes to mind is the central angle. It is the central angle's ability to sweep through an arc of 360 degrees that determines the number of degrees usually thought of as being contained by a circle.

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.


∠ QRS is not an
∠ TWV is not an
inscribed angle,
inscribed angle,
since its vertex
since its vertex
is not on the circle.
is not on the circle.
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.
m ∠ CAD
m
m ∠ BOC
m
m ∠ ACB
m ∠ ABC
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