Fundamental Ideas
Circles
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.
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Inscribed angle: In a circle, this is an angle formed by two chords with the vertex on the circle.
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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.
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Figure
2 shows examples of angles that are
not inscribed angles.
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Refer to Figure
3 and the example that accompanies it.
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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 .
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Example 2: Find
m ∠
A and
m ∠
B in Figure
5 .
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Example 3: In Figure
6 ,
QS is a diameter. Find
m ∠
R.
m ∠
R = 90°
(Theorem 72).
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Example 4: In Figure
7 of circle
O,
m
60° and
m ∠1 = 25°.
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Find each of the following.
-
m ∠ CAD
-
m
-
m ∠ BOC
-
m
-
m ∠ ACB
-
m ∠ ABC
