Name: !
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GEOMETRY: 10-2 MEASURING ANGLES AND ARCS
The two learning goals in this section are:
C.! Identify central angles, major arcs, minor arcs, and semicircles, and find their
measures.
D. !
Find arc lengths.
A Central Angle of a circle is an angle with a vertex in the center of the circle..
Its sides contain two radii of the circle.
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Sum of Central Angles
All central angles in a circle add up to 360 degrees.
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Complete Guided Practice 1 (page 706) SHOW ALL WORK!
1A. 360-165-145 = 50 degrees
1B. 360-90-40-85 = 145 degrees
Complete the table.
Arc
Definition
Measure
Minor Arc
The shortest arc
connecting two
endpoints on a
circle.
Less than 180 and
equal to the
measure of its
central angle.
Major Arc
The longest arc
connecting two
endpoints on a
circle.
Greater than 180
and equal to 360
minus the minor arc
with same
endpoints.
Diagram
Arc
Definition
Measure
Semicircle
Arc with endpoints
on a diameter.
Exactly 180
degrees.
Diagram
(Notice that the MEASURE of an arc is in degrees .)(unit)
The symbol for ʻarcʼ is
.
ABC
(NOTE: Minor Arcs: 2 Letters, Major Arcs & Semicircles: 3
Letters)
The symbol to denote ʻmeasureʼ of an arc is m.
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Complete Guided Practice 2 (page 707)
2A. minor arc, 65 deg.
2B. semicircle, 180 deg.
2C. major arc, 295 deg.
Congruent arcs are arcs ! arcs in the same or congruent circles that have the same
measure.
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Theorem 10.1
Two minor arcs are congruent if and only if !
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their central angles are congruent..
Complete Guided Practice 3 (page 708)
3A. 0.14(360) = 50.4 deg
3B. 0.14(360) = 50.4 deg
Adjacent arcs are arcs in a circle that have exactly one point in common (share a radii).
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Postulate 10.1 Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the sum of the measures of the two
arcs. (similar to Segment addition postulate and Angle addition postulate)
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Complete Guided Practice 4 (page 708)
4A.
 + DE
 = CE,
 CD
 = 180 − 63 − 90 = 27
CD
27 + 90 = 117
4B.
 = AB
 + BD
 = 117 + 90 = 207
ABD
Arc length is !distance between the endpoints along an arc measured in linear units
(fraction of the circumference).
Arc length is a portion of a circle.
Arc Length Formula:
l=
x
i2π r
360
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Complete Guided Practice 5 (page 709) SHOW YOUR WORK!
5A.!
r = 3 cm, x = 45!
l=
!
5B.!
r = 7 m, x = 80!
5C. r = 8 ft, x = 120
45
80
120
i2π (3) = 2.36cm l =
i2π (7) = 9.77m l =
i2π (8) = 16.76 ft
360
360
360