Arcs & Chords of Circles
Section 10.2
Goal:
To use properties of arcs and chords
of circles.
Central Angle
The sum of the measures of the central angles of a circle is 360°.
P
P
E
central angle - separates the circle into two arcs
A
B
AB - a minor arc of
AEB - a major arc of
P (less than 180)
P (greater than 180)
Measures of Arcs
Arcs are measured by their corresponding central angles.
If mPCM  70, then mPM  70
P
C
70
M
mPCM  mPM
E
The measure of a minor arc is always less than 180
The measure of a major arc is always greater than 180
Measures of Arcs
If the measure of an arc = 180, then you have a semicircle!
M
C
E
D
mEDM  m EDM  180
Example
A, BD is a diameter
Find : mCD, mBC , mCDB, mBCD
B
C
A 148°
D
Arc Addition Postulate
mBCD  mBC  mCD
B
C
A
D
Congruent Arcs
Congruent arcs are arcs that have the same measure and are of
the same or congruent circles.
A
B
CD 
57°
Are they congruent arcs?
57°
C
AB 
D
X
Z
XY 
65°
ZW 
Y
Are they congruent arcs?
W
Congruent Arcs
C
Y is the midpoint of XYZ if XY  YZ
X
Y
C
Z
Theorem 10.4
In the same circle or in congruent circles, two minor arcs are
congruent if and only if their corresponding chords are congruent.
A
3x  11
Q
C
B
2x  48
AB  BC if and only if AB  BC
ex. Find AB if AB  3x  11 and BC =2x+48
Theorem 10.5
If a diameter of a circle is perpendicular to a chord, then the
diameter bisects the chord and its arc.
H
D
E
G
F
If HG  DF, then DE  EF and DG  GF
Theorem 10.6
If one chord is a perpendicular bisector of another chord, then the
first chord is a diameter.
H
E
G
D
F
If DG is a perpendicular bisector of HF, then DG is a diameter.
Theorem 10.7
In the same circle or congruent circles, two chords are congruent if
and only if they are equidistant from the center.
H
D
C
B
A
G
F
DG  HF if and only if CA  CB.
Example
Given: AB  8, DE  8, CD  5
Find: CF
A
F
C
5
D
B
E
Example
Find mBD, mBED, mBE,
B
C
60°
A
82°
100°
E
D