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Other Angle Relationships
Section 10.6
Tangent-Chord Theorem
If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc.
21
B
A
Cm1 = 1
2mAB
m2 = 1
2mBCA
Example 1
m
102
T
R
S
Line m is tangent to the circle. Find mRST
mRST = 2(102 )
mRST = 204
Try This!
Line m is tangent to the circle. Find m1
m
150
1
T
Rm1 =
1
2(150 )
m1 = 75
Example 2
(9x+20)
5x
D
B
CA
BC is tangent to the circle. Find mCBD.
2(5x) = 9x + 20
10x = 9x + 20
x = 20
mCBD = 5(20 )
mCBD = 100
Interior Intersection Theorem
If two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
m1 = 1
2(mCD + mAB)
m2 = 1
2(mAD + mBC)
21
A
C
D
B
Exterior Intersection 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.
Diagrams for Exterior Intersection Theorem
1
BA
C
m1 = 1
2(mBC - mAC)
2
P
RQ
m2 = 1
2(mPQR - mPR)
3
X
W
YZ
m3 = 1
2(mXY - mWZ)
Example 3
Find the value of x.
174
106
x
P
R
Q
S
x = 1
2(mPS + mRQ)
x = 1
2(106+174 )
x = 1
2(280)
x = 140
Try This!
Find the value of x.
120
40
x
T
R
S
U
x = 1
2(mST + mRU)
x = 1
2(40+120 )
x = 1
2(160)
x = 80
Example 4
Find the value of x.
200
x 72
72 = 1
2(200 - x )
144 = 200 - x
x = 56
Example 5
Find the value of x.
mABC = 360 - 92
mABC = 268 x92
C
AB
x = 1
2(268 - 92)
x = 1
2(176)
x = 88
Chord Product Theorem
If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
E
C
D
A
B
EA EB = EC ED
Example 1
Find the value of x.
x
96
3
E
B
D
A
C3(6) = 9x
18 = 9x
x = 2
Try This!
Find the value of x.
x 9
18
12
E
B
D
A
C
9(12) = 18x
108 = 18x
x = 6
Secant-Secant Theorem
If two secant segments share the same endpoint outside a circle, then the product of the length of one secant segment and the length of its external segment equals the product of the length of the other secant segment and the length of its external segment.
C
A
B
ED
EA EB = EC ED
Secant-Tangent Theorem
If a secant segment and a tangent segment share an endpoint outside a circle, then the product of the length of the secant segment and the length of its external segment equals the square of the length of the tangent segment.
C
A
E
D
(EA)2 = EC ED
Example 2
Find the value of x.
LM LN = LO LP
9(20) = 10(10+x)
180 = 100 + 10x
80 = 10x
x = 8 x
10
11
9
O
M
N
L
P
Try This!
Find the value of x.
x
1012
11
H
GF
E
D
DE DF = DG DH
11(21) = 12(12 + x)
231 = 144 + 12x
87 = 12x
x = 7.25
Example 3
Find the value of x.
x
12
24
D
BC
A
CB2 = CD(CA)
242 = 12(12 + x)
576 = 144 + 12x
432 = 12x
x = 36
Try This!
Find the value of x.
3x5
10
Y
W
X Z
WX2 = XY(XZ)
102 = 5(5 + 3x)
100 = 25 + 15x
75 = 15x
x = 5
