Download Discuss on Central Angles and Arcs

Discuss on Central Angles and Arcs
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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.
Central angles are angles formed by any two radii in a circle. The vertex is the center of
the circle. In Figure 1, ∠ AOB is a central angle.
Figure 1 A central angle of a circle.
Arcs
An arc of a circle is a continuous portion of the circle. It consists of two endpoints and all
the points on the circle between these endpoints. The symbol is used to denote an arc.
This symbol is written over the endpoints that form the arc. There are three types of arcs:

Semicircle: an arc whose endpoints are the endpoints of a diameter. It is named
using three points. The first and third points are the endpoints of the diameter, and
the middle point is any point of the arc between the endpoints.

Minor arc: an arc that is less than a semicircle. A minor arc is named by using
only the two endpoints of the arc.

Major arc: an arc that is more than a semicircle. It is named by three points. The
first and third are the endpoints, and the middle point is any point on the arc
between the endpoints.
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In Figure 2, AC is a diameter.
is a semicircle.
Figure 2 A diameter of a circle and a semicircle.
In Figure 3,
is a minor arc of circle P.
Figure 3 A minor arc of a circle.
In Figure 4,
is a major arc of circle Q.
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Figure 4 A major arc of a circle.
Arcs are measured in three different ways. They are measured in degrees and in unit
length as follows:

Degree measure of a semicircle: This is 180°. Its unit length is half of the
circumference of the circle.

Degree measure of a minor arc: Defined as the same as the measure of its
corresponding central angle. Its unit length is a portion of the circumference. Its
length is always less than half of the circumference.

Degree measure of a major arc: This is 360° minus the degree measure of the
minor arc that has the same endpoints as the major arc. Its unit length is a portion
of the circumference and is always more than half of the circumference.
In these examples, m
indicates the degree measure of arc AB, l
length of arc AB, and
indicates the arc itself.
indicates the
Example 1: In Figure 5, circle O, with diameter AB has OB = 6 inches. Find (a) m
and (b) l
.
Figure 5 Degree measure and arc length of a semicircle.
is a semicircle. m
= 180°.
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Since
is a semicircle, its length is half of the circumference.
Postulate 18 (Arc Addition Postulate): If B is a point on
=m
, then m
.
Example 2: Use Figure 6 to find m
(m
= 60°, m
= 150°).
Figure 6 Using the Arc Addition Postulate.
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+m
Example 3: Use Figure of circle P with diameter QS to answer the following.
a. Find m
b. Find m
c. Find m
d. Find m
Figure 7 Finding degree measures of arcs.
a. m
(The degree measure of a minor arc equals the measure of its corresponding
central angle.)
b.
c. m
d. m
= 180° (
is a semicircle.)
= 130°
= 310° (
is a major arc.) The degree measure of a major arc is 360° minus
the degree measure of the minor arc that has the same endpoints as the major arc.
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The following theorems about arcs and central angles are easily proven.
Theorem 68: In a circle, if two central angles have equal measures, then their
corresponding minor arcs have equal measures.
Theorem 69: In a circle, if two minor arcs have equal measures, then their corresponding
central angles have equal measures.
Example 4: Figure 8 shows circle O with diameters AC and BD. If m ∠1 = 40°, find
each of the following.
Figure 8 A circle with two diameters and a (nondiameter) chord.
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a. m
= 40° (The measure of a minor arc equals the measure of its corresponding
central angle.)
b. m
= 40° (Since vertical angles have equal measures, m ∠1 = m ∠2. Then the
measure of a minor arc equals the measure of its corresponding central angle.)
c. m
= 140° (By Postulate 18, m
= 180°, or m
+m
=m
is a semicircle, so m
+ 40°
= 140°.)
d. m ∠ DOA = 140° (The measure of a central angle equals the measure of its
corresponding minor arc.)
e. m ∠3 = 20° (Since radii of a circle are equal, OD = OA. Since, if two sides of a triangle
are equal, then the angles opposite these sides are equal, m ∠3 = m ∠4. Since the sum of
the angles of any triangle equals 180°, m∠3 + m ∠4 + m ∠ DOA = 180°. By
replacing m ∠4 with m ∠3 and m ∠ DOA with 140°,
f. m ∠4 = 20° (As discussed above, m ∠3 = m ∠4.)
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