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.

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