12.2A Arcs and Chords
Objectives: G.C.2: Identify and describe relationships among inscribed angles, radii, and chords.
For the board: You will be able to use properties of arcs and chords of circles.
Bell Work:
1. What percent of 60 is 18?
2. What number is 44% of 6?
3. Find m<WVX.
Payment
Methods
X
V
Y
x = 30
Check
29%
W
Credit Card
30%
x/100 = 18/60
60x = 1800
0.44(6) = 2.64
29% of 360 = 104.4°
Cash
41%
Anticipatory Set:
A central angle is an angle whose vertex is the center of a circle.
An arc is an unbroken part of a circle consisting of two points called endpoints and all the points on the
circle between them.
ARC
A minor arc is an arc whose
points are on or in the interior
of a central angle.
MEASURE
DIAGRAM
The measure of a minor arc is
equal to the measure of its
central angle.
A
B
x°
P
mAB = m<APB = x°
A major arc is an arc whose
points are on or in the exterior of
a central angle.
C
The measure of a major arc is
equal to 360° - the measure of its
central angle.
A
B
x°
P
mACB = 360° - m<APB
= 360° - x°
If the endpoints of an arc lie on a
diameter the arc is a semicircle.
C
The measure of a semicircle is
equal to 180°.
mACB = 180°
A
C
P
B
Adjacent segments are collinear segments that share a common point.
Adjacent angles are angles which share a common side.
Adjacent arcs are arcs of the same circle that share exactly one point.
A
AB and BC are adjacent arcs.
B
C
P
Segment Addition Postulate
AB + BC = AC
Angle Addition Postulate
m<APB + m<BPC = m<APC
Arc Addition Postulate
The measure of an arc formed by two adjacent arcs is the sum of the
measures of the two arcs.
mABC = mAB + mBC
B
C
A
P
Read example 2 on page 803, then with your partner complete practice 2.
Practice 2: Given AC and BE are diameters, find mBD.
B
mBDE = 180° mBD + mDE = mBDE
mBD + 52° = 180°
mBD = 180° – 52° = 128°
F
A
C
93°
52°
D
E
Within a circle or congruent circles, congruent arcs are two arcs that have the same measure.
Theorem
In the same circle, or in congruent circles,
a. Congruent central angles have congruent chords.
Given: <APB  <CPD
Conclusion: AB  CD
b. Congruent chords have congruent arcs.
Given: AB  CD
Conclusion: AB  CD
c. Congruent arcs have congruent central angles.
Given: AB  CD
Conclusion: <APB  <CPD
B
A
Read example 3 on page 804, then with your partner complete practice 3.
Practice 3:
A. TV  WS. Find mWS
V
S
9a – 11 = 7a + 11
(9a – 11)°
2a = 22
a = 11
(7a + 11)°
mWS = 7(11) + 11 = 88°
T
P
W
D
P
C
B. circle C  circle J, and m<GCD = m<NJM. Find NM.
D
14t – 26 = 5t + 1
14t - 26
9t = 27
t=3
NM = 5(3) + 1 = 16
G
C
M
5t + 1
J
N
Assessment:
Student pairs will complete “CHECK IT OUT” prob. 2 and 3 from this section.
Independent Practice:
Text: pgs. 806 – 809 prob. 11-16, 25-30, 38, 39.