Lesson 9.7: Angles of Chords, Secants and Tangents
Objectives:

Find the measures of angles formed by chords, secants and tangents
Standards:
MATH.CA.8-12.GEOM.1.0, MATH.CA.8-12.GEOM.2.0, MATH.CA.8-12.GEOM.7.0, MATH.CA.8-12.GEOM.21.0
MATH.NCTM.9-12.GEOM.1.1, MATH.NCTM.9-12.GEOM.1.3
09.7.1 Theorem 9-11
F
D
9-11
A
Measure of Tangent-Chord Angle
The measure of an angle formed by a chord and a tangent
that intersect on the circle equals half the measure of the
intercepted arc.
In other words:
E
B
C
1
m FAB  mADB and
2
1
m EAB  m ACB
2
F
D
Proof:
A
Draw the radii of the circle to points A and B.
E
Triangle AOB is isosceles, therefore
O
B
1
1
BAO  ABO  (180  AOB)  90  AOB
2
2
C
We also know that, BAO  FAB  90 because FE is tangent to the circle.
We obtain, 90 
1
1
AOB  FAB  90  FAB  AOB
2
2
1
m ADB .
2
Since AOB is a central angle that corresponds to arc ADB therefore, m FAB 
This completes the proof.
Example 1: Find the values of a, b and c.
First we find angle a: 50  45  a  180   a  85
Using Theorem 9-11 we conclude that:
o
m AB =2(50 ) = 100
and
a 45o
o
m AC =2(45o) = 90o
Therefore,
A
50o
1
b  90  45
2
1
c  100  50
2
B
b
c
C
09.7.2 Theorem 9-12
9-12
A
B
Angles Inside a Circle
a
D
The measure of the angle formed by two chords that intersect inside a
circle is equal to half the sum of the measure of the intercepted arcs.
In other words: a 
C
1
m AB  mDC 
2
Proof:
Draw a line to connect points B and C.
1
mDC
2
1
ACB  mAB
2
a  ACB  DBC
DBC 
a 
1
1
mDC  m AB
2
2
a 
1
(mDC  m AB )
2
A
B
Inscribed angle
a
D
Inscribed angle
C
Alternate angle theorem
Substitution
This completes the proof.
A
Example 2:
40
Find the value of angle x.
D
Solution:
a 
B
o
x
1
mAD  mBC   1 40  62  51
2
2
62o
C
09.7.3 Theorem 9-13
9-13
Angles Outside a Circle
a
x
The measure of an angle formed by two secants drawn from a point
outside the circle is equal to half the difference of the measures of
the intercepted arcs.
In other words: a 
yo
1
( y  x)
2
o
This theorem also applies for an angle formed by two tangents to the circle drawn from a point outside the
circle and for an angle formed by a tangent and a secant drawn from a point outside the circle.
a
a
x
o
xo
yo
yo
Proof:
Draw a line to connect points A and B.
1
DBA  x
2
1
BAC  y
2
BAC  DBA  a
1
1
y  x  a
2
2
a 
Inscribed angle
D
a
xo
B
A
Inscribed angle
yo
Alternate angle theorem
C
Substitution
1
( y  x)
2
This completes the proof.
Example 3:
Find the value of angle x.
54o
Solution:
1
x  220   54  83
2
220o
x
Homework:
Find the value of the missing variables.
1.
2.
65o
3.
200o
y
x
22o
64o
z
140o
4.
120o
5.
6.
(2z – 15)o
v
u
(z + 15)o
35o
60o
300
45o
30o
7.
8.
9.
y
(8z - 20)o
25o
40o
75o
x
10.
11.
a
c b
12.
40o
42o
82o
60o
13.
2x
a
x
b
c
14.
15.
55
o
y
100o
35o
x
y
60o
x
200o
x
16.
17.
18.
40o
146o
x
56o
80o
40o
x
x
19. Find the following angles
C
a) OAB
B
E
b) COD
c) CBD
O
D
d) DCO
35o
e) AOB
45o
A
f) DOA
20. Find the following angles
D
a) CDE
b) BOC
c) EBO
E
A
O
55o
C
d) BAC
B
21. Four points on a circle divide it into four arcs, whose sizes are 44 degrees, 100 degrees, 106 degrees,
and 110 degrees, in consecutive order. The four points determine two intersecting chords. Find the sizes of
the angles formed by the intersecting chords.
Answer:
1. 102.5o
2. 21o
3. 100o
4. 40o
5. 90o
6. 90o
7. 30o
8. 25o
9. 210o
10. a = 60o, b = 80o, c = 40o
11. a = 82o, b = 56o, c = 42o
12. 45o
13. 55o
14. x = 60o, y = 105o
15. 20o
16. 50o
17. 60o
18. 45o
19. a. 45o
b. 80o
c. 40o
d. 50o
20. a. 90o
b. 110o
c. 55o
d. 20o
21. 75o and 105o
e. 90o
f. 110o