Angles Related to a Circle

Section 10.5

Works Cited:By: Tara Mazurczyk

“Geometry.” Glencoe. 19 May 2008 <http://www.glencoe.com/ sec/ math/ geometry/ geo/ geo_04/ extra_examples/ chapter10/ extras_10_6.rtf>.McDougal, Littell & Company. “Angles Related To a Circle.” Geometry for Enjoyment and Challenge. Evanston: n.p., 1991.

468-78.Roberts. “Formulas for Working with Angles in Circles.” http://regentsprep.org. 19 May 2008 <http://regentsprep.org/ REgents/ mathb/ 5A1/ CircleAngles.htm>.

1.  Central Angle:  A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle.

Central Angle = Intercepted Arc <AOB is a central angle.

 Its intercepted arc is the minor arc from A to B.  m<AOB = 80º

 

The measure of an inscribed angle or a tangent-chord angle is ½ the measure

of its intercepted arc

A

D

B

C

50

ADC is a central angle

Example:

A

D

B

C

70Given: mAC =

Find: ABCm

mACABCm2

1

35

70*2

1

Inscribed Angles

70

An inscribed angle is an angle whose vertex is on a circle and whose sides are determined by two chords.

mACADCm

mAC2

1ABCm

Tangent-chord Angles

D

F

E

DE is a tangent and EF is a chord

DEF is a tangent-chord angle

A tangent-chord angle is an angle who vertex is on a circle and whose sides are determined by a tangent and a chord that intersect at the tangent’s point of contact

Example:

A

C

B

Given: AB is tangent at B,

Find:

90mBC

ABCm

mBCABCm2

1

45

90*2

1

Tangent-tangent Angles

A

B

C

AB and BC are tangents which means that 180ABCmmAC

Example:

C

D

E

Given:

Find:

60

60mCE

120

18060

180

Dm

Dm

DmmCE

Dm

A tangent-tangent angle is an angle whose vertex is outside a circle and whose

sides are determined by two tangents

mACmADCBm 2

1

D

The measure of a chord-chord angle is ½ the sum of the measures of the arcs

intercepted by the chord-chord angle and its vertical angle.

Chord-chord Angles

A chord-chord angle is an angle formed by two chords that intersect inside a circle but not at the center.

A B

C

D

AC and BD are chords

E

Example:A B

C

D

E

Given:

Find:

mCDmABAEBm 2

1

30

70

mCD

mAB

AEBm

50

1002

1

30702

12

1

AEBm

AEBm

AEBm

mCDmABAEBm

The measure of a secant-secant angle, a secant-tangent angle, or a tangent-tangent angle is ½ the difference of the measures

of the intercepted arcs.

Secant-secant Angles

A secant-secant angle is an angle whose vertex is outside a circle and whose sides are determined by two secants.

A

B

C

D

E

AC and CD are secants

mBDmAECm 2

1

Example:

A

B

C

D

E

Given:

Find:

40

120

mBD

mAE

Cm

40

802

1

401202

12

1

Cm

Cm

Cm

mBDmAECm

Secant-tangent Angles

A secant-tangent angle is an angle whose vertex is outside a circle and whose sides are determined by a secant and a tangent.

A

B

C

D

AB is a tangent and BD is a secant

mACmADBm 2

1

Example:

Given:

140

80

mCD

mAC

Bm

30

602

1

801402

12

1

140

80140360

Bm

Bm

Bm

mACmADBm

mAD

mAD

B

C

D

A

Find:

Practice Problems:

1.) A B

C

D

E

Given:

Find:

140

60

mAD

AEBm

mBC

2.) A

B

C

D

Given:

Find:

260mADC

BmAB and BC are tangent to EE

3.) Given:140mAC

BmFind:

A

C

DB

4.)

150

50

mCD

mAC

Bm

C

D

A

Find:

Given:

B

5.)

140

80

40

50:

EF

CD

BC

ABGiven

A

BC

D

E

F

H

G

50

40

80

140

12

3

4

Find:1m2m3m4m

mAF

mDE

FG is tangent to H

6.)

A

B

C

D

G

F

E

I

H

J

K

Given:

70

130

80

120

140

mGF

mEL

mHG

mBH

mCI

140

120

80

130

70

L

AB and AH are tangent to J

6

5

4

3

2

1

m

m

m

m

m

m

Find:

1

23

4

5

6

Answers:

1.)

100

140240

1402

1120

2

1

120

18060

180

mBC

mBC

mBC

mBCmADBECm

BECm

BECm

AEBmBECm

80

1602

1

1002602

12

1

100

260360

Bm

Bm

Bm

mACmAECBm

mAC

mAC2.)

3.)

40

802

1

1402202

12

1

220

140360

Bm

Bm

Bm

mACmADCBm

mADC

mADC 4.)

55

1102

1

501602

12

1

160

50150360

Bm

Bm

Bm

mACmADBm

mAD

mAD

5.)

10

40

204

653

452

201

DE

AF

m

m

m

m 6.)

306

855

204

603

502

401

m

m

m

m

m

m