CENTRAL AND INSCRIBED ANGLES
#18
A central angle is an angle whose vertex is the center of a circle
and whose sides intersect the circle. The degree measure of a
central angle is equal to the degree measure of its intercepted arc.
For the circle at right with center C, ∠ACB is a central angle.
An INSCRIBED ANGLE is an angle with its vertex on the circle
and whose sides intersect the circle. The arc formed by the
intersection of the two sides of the angle and the circle is called an
! is an
INTERCEPTED ARC. ∠ADB is an inscribed angle, AB
intercepted arc.
The INSCRIBED ANGLE THEOREM says that the measure of
any inscribed angle is half the measure of its intercepted arc.
Likewise, any intercepted arc is twice the measure of any
inscribed angle whose sides pass through the endpoints of the arc.
m∠ADB =
1
2
A
B
C
A
•
• B
•
D
! = 2m∠ADB
! and AB
AB
Example 1
Example 2
! and
P, m !CPQ = 70˚ . Find mCQ
!.
mCRQ
In the circle shown below, the vertex of
the angles is at the center. Find x.
In
C
B
2x
R
P
Q
C
! = m !CPQ = 70˚
mCQ
! = 360° ! mCQ
" = 360° = 70° = 290°
mCRQ
A
x
Since AB is the diameter of the circle,
x + 2x = 180˚ ⇒ 3x = 180˚ ⇒ x = 60˚
Example 3
Solve for x.
˚
100
A
B
x
C
! = 220˚, mBC
! = 360˚ – 220˚ = 140˚. The
Since mBAC
! or 70˚.
m!BAC equals half the mBC
120˚
GEOMETRY Connections
© 2007 CPM Educational Program. All rights reserved.
51
P if m!WPX = 28˚ , m!ZPY = 38˚ , and WZ and XV are
Find each measure in
diameters.
!
1. YZ
5.
!XPY
!
2. WX
6.
!
XY
3. !VPZ
7.
!
XWY
!
4. VWX
8.
!
WZX
W
P
V
X
Z
Y
In each of the following figures, O is the center of the circle. Calculate the values of x and
justify your answer.
9.
10.
O
11.
136°
O
x
146°
O
49°
x
13.
14.
12.
O
O
•
O
18°
x
x
16.
150°
x
O
62°
15.
100°
O
x
•
x
x
27°
100°
17.
18.
55°
O
•
19.
12°
x
O
20.
O
•
•
35° x
91°
x
x
O
•
29° 101°
Answers
1. 38˚
2. 28˚
3. 28˚
4. 180˚
5. 114˚
6. 114˚
7. 246˚
8. 332˚
9. 68˚
10. 73˚
11. 98˚
12. 124˚
13. 50˚
14. 55˚
15. 18˚
16. 27˚
17. 55˚
18. 77˚
19. 35˚
20. 50˚
52
© 2007 CPM Educational Program. All rights reserved.
Extra Practice