Geometry Semester 2
Homework on Arcs and Angles
1.
Name: ___________________________
In a triangle, an exterior angle is the sum of the two remote interior angles. In order to see why
this is true, consider the following with exterior angle x and remote interior angles a and b:
b
a
2.
c
x
(a)
In this figure, c  x  180 . Why? ________________________________________
(b)
In the triangle, a  b  c  180 . Why? _____________________________________
(c)
So c  x  a  b  c . Solve this equation for x:
x = _____________
P
Now we will apply that result to find the measure
of an angle between two chords of a circle:
a
x
Q
T
O
R
S
b
(a) In the above figure, x is an exterior angle of triangle PST, so it equals the sum of which
two angles? x = ________ + _________
1
a . Likewise, SPR is
2
1
an inscribed angle, so it is also half of its intercepted arc. That is, SPR 
________.
2
(b) PSQ is an inscribed angle, so it is half of arc a: PSQ 
(c) Using the results from parts (a) and (b), what is x in terms of arcs a and b?
x = ________________
Page 1 of 5
3.
Now we will apply the same idea when the vertex
of the angle is outside the circle.
P
R
O
a
x
b
T
S
Q
(a) In the above figure, PRQ is an exterior angle of  QRT so PRQ  x   _______.
(b) But PRQ 
(c) So
1
a . Why? ___________________________
2
1
1
a  x  RQT . But RQT is the same angle as RQS so it is ______
2
2
(d) So we now have
1
1
a  x  b . Solve this for x: x = _________________
2
2
If one or both of the lines are tangent to the circle, the same result holds. In the following
ab
figure, x 
:
2
P
R
x
a
O
T
b
Q
tangent
4.
20
Find z in the figure on the right:
140
o
z
Page 2 of 5
o
Problems 5-8: Complete the definition and theorems about angles and arc measures.
5.
Definition of Arc Measure:
mAB = m ___________
A
mACB = _________________
6.
B
Measure of Inscribed Angle:
mADB 
O
7.
Angle inscribed in a Semicircle:
mADC  ____ degrees
C
8.
Two Inscribed Angles that intercept the Same Arc:
D
ADB   ________
Problems 9-12: Find the measures of the angles, using the given arc measures.
Measure of Angle Between Two Chords:
A
10.
C
o
23
Q
Measure of Angle Between Two Secants:
G
D
48
O
mGPF 
65
11.
12.
o
E
Measure of Angle Between a Secant and Tangent:
F
mBPF 
Measure of Angle Between Two Tangents:
mBPE 
Page 3 of 5
o
ne
61
53
P
o
nt
li
mAQC 
tangent line
o
28 B
83
o
o
tan
ge
9.
13.
What is the measure of arc ABC ?
What is the measure of G ?
14.
B
70 
30 
C
A
G
80
D
15.
What is the measure of MQN ?
16.
What is the measure of PQ ?
34 
130 
P
Q
M
40 
80
Q
(tangent line)
N
17.
If mBQ = 36o and mQD = 80o,
B
then what is mP  mQ  ?
A
36
P
o
Q
D
80
o
C
18.
What is the measure of any angle inscribed
?
in a semicircle?
A
19.
If AB is a diameter, BD is a tangent,
and ABC measures 40o , what is the
C
measure of D ?
B
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D
20. What fraction of the circle on the right is shaded?
21. What fraction of this circle is shaded?
150
o
22. If the circle in problem 21 has a radius of 10 cm, then what is the area of the shaded region?
23. If a 120o arc is 6 cm long, then what is the circumference of the whole circle? __________
24. If the circumference of the circle on the right is 62.8 cm, then what
120
is the area of the shaded region?
25.
In an arcade game, the “monster” is the shaded
sector of a circle of radius 1 cm, as shown in
the figure. The missing piece (the mouth) has a
central angle of 60o. What is the area of the
monster?
Page 5 of 5
m
1c
o
60
o