Lesson 9 - High Tech High
Lesson 9.7: Angles of Chords, Secants and Tangents
1
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
9-11 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: m[pic] and
m[pic]
Proof:
Draw the radii of the circle to points A and B.
Triangle AOB is isosceles, therefore
[pic]
We also know that, [pic]because FE is tangent to the circle.
We obtain, [pic]
Since[pic]is a central angle that corresponds to arc therefore, m[pic] .
This completes the proof.
Example 1: Find the values of a, b and c.
First we find angle a: [pic]
Using Theorem 9-11 we conclude that:
m =2(50o) = 100o
and m =2(45o) = 90o
Therefore, [pic]
[pic]
09.7.2 Theorem 9-12
9-12 Angles Inside a Circle
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: [pic]
Proof:
Draw a line to connect points B and C.
[pic] Inscribed angle
[pic] Inscribed angle
[pic] Alternate angle theorem
[pic] Substitution
[pic]
This completes the proof.
Example 2:
Find the value of angle x.
Solution:
[pic]
09.7.3 Theorem 9-13
9-13 Angles Outside a Circle
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: [pic]
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.
Proof:
Draw a line to connect points A and B.
[pic] Inscribed angle
[pic] Inscribed angle
[pic] Alternate angle theorem
[pic] Substitution
[pic]
This completes the proof.
Example 3:
Find the value of angle x.
Solution:
[pic]
Homework:
Find the value of the missing variables.
1. 2. 3.
4. 5. 6.
7. 8. 9.
10. 11. 12.
13. 14. 15.
16. 17. 18.
19. Find the following angles
a) [pic]
b) [pic]
c) [pic]
d) [pic]
e) [pic]
f) [pic]
20. Find the following angles
a) [pic]
b) [pic]
c) [pic]
d) [pic]
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 e. 90o f. 110o
20. a. 90o b. 110o c. 55o d. 20o
21. 75o and 105o
-----------------------
(z + 15)o
v
60o
120o
u
45o
ACB
35o
F
x
65o
140o
(2z – 15)o
300
F
E
ADB
200o
z
64o
22o
y
E
x
30o
D
C
y
25o
B
75o
(8z - 20)o
A
40o
D
C
B
A
O
ADB
ADB
a
b
c
C
B
A
50o
45o
AB
AC
a
A
B
C
D
AB
DC
D
C
B
A
a
DC
AB
DC
AB
DC
AB
a
xo
62o
D
C
B
yo
yo
A
yo
xo
a
xo
yo
a
xo
a
D
C
B
x
A
40o
AD
BC
b
40o
220o
54o
c
c
60o
a
x
82o
a
b
42o
x
35o
2x
y
x
y
x
100o
55o
60o
x
200o
x
80o
x
40o
40o
x
56o
146o
E
O
45o
35o
A
B
C
D
E
D
C
B
A
O
55o
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