Angles on Circle

Name _________________________________Period____
Date_______
Background Information – You will need to use this information to complete the worksheet that
follows.
Central angle – an angle formed by radii of a circle.
O
A
O is the central angle subtended by arc AB
Inscribed angle – an angle formed by connecting two points on the circumference of a circle to
another point on the circumference.
B
A
B
C is an inscribed angle subtended by arc AB
C
Cyclic Quadrilateral – It is a quadrilateral that is formed by four chords
Angle Properties:
1. Angles in a Circle Theorem
The measure of the central angle is twice the measure of an inscribed angle subtended(opposite
to) by the same arc.
A
B
O
C
O = 2C
Both are subtended by arc AB

2. The angle inscribed in a semicircle is a right angle.
A
C
O
B C = 90°
3. Inscribed angles subtended(opposite to) by the same arc are equal.
D
C
B
C = D
Both are subtended by arc AB
A
4. The sum of opposite angle on cyclic quadrilateral is 180 0 .
A andC are opposite angles so
B andD are opposite angles so
A +C=180 0 .
Practice:
Part 1 : Central & Inscribed Angles Subtended by Arcs
Use this drawing to answer the questions below.
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 made by.
4. AOB is made by arc _______.
5. FDG is made by arc _______.
6. ACB is made by arc _______.
For questions 7 – 9, fill in the missing angles.
7. The only central angle made by arc AB is __________
8. The two inscribed angles are made by the arc AB are __________ and __________
9. The angle made by arc FG is __________. It is a(n) ________________ (central / inscribed)
angle.
B +D=180 0 .
F
E
G
O
A
D
C
B

Part 2 : Finding Missing Angles using Angles in a Circle Theorem
Use the diagrams to complete the sentence(s) following each one.
1. ACB is an inscribed angle made by arc ______. The central angle made by the same arc is
__________. Therefore, the measurement of AOB is ______°.
A
O
37°
B
C
2. ROT is a central angle made by arc ______. An inscribed angle made by the same arc is
__________. Therefore, the measure of RST is ______°.
R
86°
O
T
S
3. Reflex FOH is the central angle made by major arc ______. An inscribed angle made by the
same arc is _______. Therefore, the measure of FGH is _____°.
F
G
216°
O
H

4. RPQ is the inscribed angle made by major arc ______. _______ is also made by this
same arc, and its measure is _______°. Therefore, the measure of RPQ
is ______°.
R
P
70°
O
55°
Q
5. Since the sum of angles in a quadrilateral is ______°, the measure of PRO is ______°.
Part 3 : 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 ______°.
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_____°.
O
A
C
B

Part 4 : Finding Missing Angles using Inscribed Angles Subtended by the Same Arc
Use the diagrams to complete the sentence(s) following each one.
1. DAC is subtended by arc _____. Another inscribed angle subtended by the same arc is
________. Therefore, the measure of DBC is ______°.
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 ______°.
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 _______°.

Part 5 : Cyclic Quadrilateral
1. Look at the following figure!
2. Look at the following figure!
If PQR = 125 0 , QRS = 78 0 , find the measure of:
a. SPQ
b. RSP
3. Look at the following figure!
If BAD = x + 20, BCD = 3x, find the measure of:
a. BOD minor
b. BOD major
4. Look at the following figure!
Find the value of x and y!

Part 6 : 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.