Internal & External Angles of Circles
1
Exterior Angle (Out)
Interior Angle (In)
Inscribed Angle (On)
ANGLES ON THE CIRCLE
Theorem – If a tangent and a secant (chord) intersect at a point on a circle, then
the measure of each angle formed is one half the measure of its intercepted arc.
m1  _________
D
R
m2  __________
G
2
1
B
Find m1.
Find x and y.
Find x and m1.
x
x
y
78°
112°
1
1
118°
Internal & External Angles of Circles
2
ANGLES IN THE CIRCLE
C
A
E
D
D
F
B
x°
C
A
x°
Theorem – If two chords intersect in the interior of a circle, then the measures of
each angle is one half the sum of the measures of the arcs intercepted by the angle
and its vertical angle.
mAEB  mCED  __________
C
A
E
D
F
B
Examples
Find x.
Find x.
Find x.
82°
55°
92°
x
71°
x
x
107°
146°
Internal & External Angles of Circles
3
ANGLES OUTSIDE THE CIRCLE
Angles can be formed outside the circle in a number of different ways.
Two Secants
One Secant and One Tangent
Two Tangents
A
Theorem – If a tangent and a secant, two tangents or two secants intersect in
the exterior of a circle, then the measure of the angle formed is one half the
difference of the measures of the intercepted arcs.
C
E
D
mAEB  ________________
B
Examples
Find x.
Find x.
Find y and x.
183°
168°
x
29°
93°
59°
x
y
x
93°
Solve for the missing values.
1
64°
a) m1 = _________
b) m2 = _________
c) m3 = _________
d) m4 = _________
3
4
5
6
33°
2
e) m5 = _________
f) m6 = _________
96°
Internal & External Angles of Circles
4
NAME: ______________________________________
1. Determine the requested value(s). (Lines that appear to be tangent are tangent.)
a)
b)
c)
d)
38°
1
143°
132°
1
92°
1
95°
1
53°
116°
104°
37°
m1 = _________
e)
m1 = _________
f)
m1 = _________
m1 = _________
g)
h)
9x - 10
128°
1
116°
132°
1
1
133°
81°
5x + 8
128°
m1 = ______
i)
x = ________
m1 = _________
m1 = ______
j)
k)
l)
2
50°
1
102°
2
1
79°
1
m1 = ___ m2 = ___
m)
1
2
m1 = ____ m2 = ____
n)
2
72°
m1 = ____ m2 = ____
o)
m1 = ____ m2 = ___
p)
160°
168°
132°
1
48°
1
51° 1
158°
78°
1
128°
m1 = _________
q)
m1 = _________
r)
m1 = _________
s)
m1 = _________
t)
149°
39°
134°
1
39°
m1 = ________
1
28°
1
m1 = ________
1
62°
43°
m1 = ________
28°
m1 = ________
Internal & External Angles of Circles
EXIT TICKET
5
NAME: _________________________
1. Given the two tangent lines, explain how you would
solve for arc 1 and angle 2.
2
1
232°
2. Use the previous question to help you solve for arc 1 and arc 2 in this
problem, show your work.
64°
2
1
Internal & External Angles of Circles
6
HOMEWORK
1. Determine the requested value(s). (Lines that appear to be tangent are tangent.)
a)
b)
c)
266°
2
63°
1
75°
126°
2
1
46°
1
m1 = ______ m2 = ______
d)
m1 = ______
m1 = _________
m2 = ______
e)
f)
94°
168°
66°
50°
1
128°
1
21°
2
2
1
m1 = _________
m1 = ______ m2 =
______
g)
h)
m1 = ______ m2 = ______
i)
2
58°
32°
55°
1
2
1
65°
1
106°
m1 = ______
m1 = ______ m2 =
______
m1 = ______
m2 = ______
Internal & External Angles of Circles
7
2. Determine the requested value(s). (Lines that appear to be tangent are tangent.)
a)
b)
c)
B
D
5x + 8
7x - 14
73°
56°
4x + 6
2x + 34
E
59°
C
A
x = ______
x = __________
mABC = _______
x = __________
3. Determine the requested value(s). (Lines that appear to be tangent are tangent.)
a)
b)
c)
78°
134°
113°
1
2
1
2 63°
42°
1
m1 = ______ m2 = ______
d)
m1 = ______ m2 = ______
e)
m1 = ______
f)
28°
1
2
1
2
58°
40°
44°
2
m1 = ______
m2 = ______
m1 = ______ m2 = ______
1
m1 = ______ m2 = ______
Internal & External Angles of Circles
g)
8
h)
i)
73°
2
1
160°
75°
1
2
35°
28°
2
18°
134°
1
m1 = ______ m2 = ______
90°
m1 = ______ m2 = ______
m1 = ______ m2 = ______
4. Determine the requested value(s). (Lines that appear to be tangent are tangent.)
a)
b)
c)
F
260°
10x - 6
5x + 24
A
228°
D
3x
8x
32°
E
x = __________
x = __________
x = __________
5. Solve for the missing values.
a) m1 = _________
b) m2 = _________
c) m3 = _________
d) m4 = _________
e) m5 = _________
f) m6 = _________
mEF = _________
100°
3
6
4
30°
1
2
6. Solve for the missing values.
3
a) m1 = _________
b) m2 = _________
c) m3 = _________
d) m4 = _________
139°
4
e) m5 = _________
f) m6 = _________
2
1
B
C
6
5
72.5°
5