Circles Worksheet
Central angle – an angle formed by radii of a circle.
A
O
 O is the central angle subtended by arc AB
B
Inscribed angle – an angle formed by connecting two points on the circumference of a circle to
another point on the circumference.
B
A
 C is an inscribed angle subtended by arc AB

C
Angle Properties:
1. Angles in a Circle Theorem
The measure of the central angle is twice the measure of an inscribed angle subtended by the
same arc.
C
A
 O = 2 C
Both are subtended by arc AB
O

B
2. The angle inscribed in a semicircle is a right angle.
B
O

A
 C = 90°
C
3. Inscribed angles subtended by the same arc are equal.
D
C

A
B
C= D
Both are subtended by arc AB
Central & Inscribed Angles Subtended by Arcs
Use this drawing to answer the questions below.
D
C
E
F
O

B
G
A
For questions 1 – 3, fill in the blank with “central” or “inscribed”.
1.  FDG is a(n) _______________ angle.
2.  AEB is a(n) _______________ angle.
3.  AOB is a(n) _______________ angle.
For questions 4 – 6, indicate which angle each arc is subtended by.
4.  AOB is subtended by arc _______.
5.  FDG is subtended by arc _______.
6.  ACB is subtended by arc _______.
For questions 7 – 9, fill in the missing angles.
7. The only central angle subtended by arc AB is __________
8. The two inscribed angles are subtended by the arc AB are __________ and __________
9. The angle subtended by arc FG is __________. It is a(n) ________________ (central /
inscribed) angle.
Use this drawing to answer the questions below.
S
(Recall: A reflex angle is an angle between
180° and 360°.)

O
P
Q
R
10. Minor arc PQ subtends the inscribed angle __________.
11. Darken major arc PQ on the diagram above. Major arc PQ subtends reflex  POQ. This
angle is indicated on the diagram. The inscribed angle subtended by major arc PQ is
_________.
Using the Angle Properties
Use this drawing to answer the questions below.
L
O
K

J
1.
2.
3.
4.
M
The central angle subtended by arc JM is __________.
An inscribed angle subtended by the same arc is ____________.
Another inscribed angle subtended by the same arc is ___________.
Use angle properties 1 and 3 to explain the relationship between the measurements of the
following angles:
a)  JOM and  JLM
b)  JOM and  JKM
c)  JLM and  JKM
Use this drawing to answer the questions below.
A
O
B
D
C
1. Which line segment in the diagram is a diameter? ______
2. According to angle property 2, the angle in the diagram that is a right angle is ___________.
It is subtended by arc ______.
3. According to angle property 3, which angle in the diagram is equal to  ADB?
Finding Missing Angles using Angles in a Circle Theorem
Use the diagrams to complete the sentence(s) following each one.
B
A
O

37°
C
1.  ACB is an inscribed angle subtended by arc ______. The central angle subtended by the
same arc is __________. Therefore, the measurement of  AOB is ______°.
R
86°
O
T
S
2.  ROT is a central angle subtended by arc ______. An inscribed angle subtended by the
same arc is __________. Therefore, the measure of  RST is ______°.
F
G
216°  O
H
3. Reflex  FOH is the central angle subtended by major arc ______. An inscribed angle
subtended by the same arc is __________. Therefore, the measure of  FGH is _____°.
P
R
55°
Q
70°

O
4.  RPQ is the inscribed angle subtended by major arc ______. Reflex  _______ is also
subtended by this same arc, and its measure is _______°. Therefore, the measure of  RPQ
is ______°.
5. Since the sum of angles in a quadrilateral is ______°, the measure of  PRO is ______°.
Finding Missing Angles using Angles Inscribed in a Semicircle
Use the diagrams to complete the sentence(s) following each one.
J
O
K
L
1. The angle inscribed in a semicircle in the above diagram is _________. Therefore the
measure of this angle is _____°.
Q
P
39°

O
R
2. Since it is inscribed in a semicircle, the measure of  PQR is ______°.
3. Since the sum of angles in a triangle is ______°, the measure of  PRQ is ______°.
A
O

C
B
4. The angle inscribed in a semicircle is _________. Its measure is _____°.
5.  ABC is an isosceles triangle. This means that  ______ and  ______ are equal.
Because the sum of angles in a triangle is 180°, this means that both of these angles measure
_____°.
Finding Missing Angles using Inscribed Angles Subtended by the Same Arc
Use the diagrams to complete the sentence(s) following each one.
A
43°
B
D
C
1.  DAC is subtended by arc _____. Another inscribed angle subtended by the same arc is
________. Therefore the measure of  DBC is ______°.
R
U
74°
21°
T
S
2.  RSU is subtended by arc ______. Another inscribed angle subtended by the same arc is
________. Therefore the measure of  RTU is ______°.
3. Another inscribed angle subtended by the same arc as  SUT is __________. Therefore the
measure of this angle is ______°.
W
25°
X
V
Z
18°
Y
4. By the inscribed angle property, the measure of  WXZ is ______°, and the measure of
 XZY is ______°.
5. Since the sum of angles in a triangle is 180°, the measure of  WVX is _______°.
Putting it All Together
In each of the following circles, use the angle properties to find the missing angles.
1.
2.
3.
4.
In each of the following circles, use the angle properties to find the missing angles. Justify your
statements with reasons.
5.
6.
7.
8.