Section 10.2: Angles and Arcs
1. I can define and identify major arcs, minor arcs,
semicircles, and central angles and find their measures.
2. I can determine arc length of a circle.
Angles and Arcs
Central angle - an angle that intersects a circle in two
points and has its vertex at the center. The sides of
the central angle contain two radii of the circle.
• The sum of the measures
of the central angles of a
circle is 360 degrees.
A
B
• A central angle separates
the circle into parts, each
of which is an arc.
• Arc - a part or portion of a
circle that is defined by two
endpoints.
c
Arcs
Arcs are usually denoted by this symbol_______.
Arcs of a Circle: Graphic Organizer
Type of Arc:
A
minor arc - an arc with a
measure of less than 180
degrees.
C
P
A
Major Arc
Semicircle
D
J
E
C
60
110
M
G
B
central angle
B
major arc - an arc
with a measure of
more than 180
degrees.
Minor Arc
Example:
The measure of each arc is related to
the measure of its central angle.
Arc Degree Measure Equals
By the letters of the two endpoints and another point on the arc. Ex:
Usually by the letters of the two endpoints. Ex: By the letters of the two endpoints and another point on the arc. Ex: The measure of the 360 minus the measure 360 ÷2 or 180 degrees central angle. Less of the minor arc. than 180 degrees.
Greater than 180 degrees. Congruent Arcs
A
Theorem 10.1:
In the same or in congruent circles, two arcs are
congruent if and only if their corresponding angles are
congruent.
But I forget
how to determine
if two circles are
congruent?
Review Examples:
Finding the Measures of Central Angles & Arcs
K
L
F
Named:
N
D
B
45
45
C
Class, help
Georgie out!
60
60
Are these arcs
congruent?
Find the measure of the following
Central Angles & Arcs :
D
O
A
45
E
20
C
B
a. Angle AOD ______________
b. Arc AD _________________
c. Angle DOC _______________
d. Angle AOB _______________
e. Arc ADE _________________
1
Examples: Finding the Measures of Arcs
Arc Addition Postulate:
Arcs of a circle that have exactly one point in
common are adjacent arcs. The measure of
adjacent arcs can be added.
C
A
O
Find the measure of each arc:
B
Postulate 10.1:
C
A
Arc AB ____________
Arc Addition Postulate: The
measure of an arc formed by two
adjacent arcs is the sum of the
measure of the two arcs.
__________________________
40
E
Arc ABD ___________
80
110
B
Arc AD _____________
D
Arc Length
Examples: Using Proportions to Find Arc Length
A
• Another way to measure an arc is by its length.
arc length - The distance along the curved line making
up the arc.
D
C
• An arc is part of the circle which means its length
is part of the circumference of the circle.
15
Find the length of each arc for the
given angle measure.
H
120
B
G
C
E
a. Arc DE if m DCE = 100
Link to Interactive
• To find arc length I can use a proportion or a formula.
degree measure of arc
degree measure of whole circle
A =
360
l
2 πr
arc length
circumference
______________________________
F
A = l
b. Arc HDF if m HCF = 125
360 2πr
______________________________
• Expressed as a formula
AxC=l
360
A = degree measure of arc
C = circumference
l = arc length
AxC=l
360
Use the
formula
this time.
C. Arc HD if m DCH = 45
______________________________
2