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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 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.
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