Download Section 10.5

Angles Related to a Circle
Section 10.5
By: Tara Mazurczyk
Works Cited:
“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
Inscribed Angles
An inscribed angle is an angle whose
vertex is on a circle and whose sides are
determined by two chords.
Example:
B
A
70
D
B
C
A
50
D
C
Given: mAC = 70
Find: mABC
ADC is a central angle
mADC  mAC
1
mABC  mAC 
2
1
mAC 
2
1
 * 70
2
 35
mABC 
Tangent-chord
Angles
Example:
B
E
A
C
D
F
DEF
is a tangent-chord angle
Given: AB is tangent at B, mBC  90
Find: mABC
DE is a tangent and EF is a chord
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
mABC 
1
mBC 
2
1
* 90
2
 45

Tangent-tangent
Angles
A tangent-tangent angle is an angle
whose vertex is outside a circle and whose
sides are determined by two tangents
Example:
C
60
A
B
E
D
Given: mCE  60
C
AB and BC are tangents which means
that mAC  mABC  180
1
mB  mADC  mAC 
2
Find: mD
mCE  mD  180
60  mD  180
mD  120
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
Example:
A
B
E
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
Given: mAB  70
mCD  30
E
Find: mAEB
C
D
AC and BD are chords
1
mAEB  mAB  mCD
2
1
mAB  mCD
2
1
mAEB  70  30
2
1
mAEB  100
2
mAEB  50
mAEB 
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
Example:
A
B
C
A secant-secant angle is an angle
whose vertex is outside a circle and
whose sides are determined by two
secants.
D
E
Given: mAE  120
mBD  40
A
B
C
D
E
AC and CD are secants
1
mC  mAE  mBD 
2
Find:
mC
1
mAE  mBD 
2
1
mC  120  40
2
1
mC  80
2
mC  40
mC 
Secant-tangent
Angles
Example:
A
Given:
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
1
mB  mAD  mAC 
2
mAC  80
mCD  140
Find: mB
B
C
D
mAD  360  140  80
mAD  140
1
mB  mAD  mAC 
2
1
mB  140  80
2
1
mB  60
2
mB  30
Practice Problems:
1.)
A
B
Given: mAEB  60
mAD  140
E
Find:
C
mBC
D
A
2.)
Given: mADC  260
B
D
AB and BC are tangent to
E
Find:
C
mB
E
3.)
A
Given:
mAC  140
B
D
Find: mB
C
4.)
A
Given:
mAC  50
mCD  150
Find: mB
B
C
D
40
5.)
Given : AB  50
BC  40
CD  80
80
EF  140
FG is tangent
to H
C
B
3
50
D
2
1
E
H
Find:
m1
m2
140
m3
4
m4
mAF
G
mDE
F
A
6.)
Given: mCI  140
B
C
A
3
120
2
80
J
140
mBH  120
mHG  80
mEL  130
mGF  70
G
4
H
70
5
Find:
m1
m2
m3
m4
m5
m6
AB and AH are
tangent to
J
K
I
F
L
1
6
E
D
130
Answers:
1.)
mBEC  mAEB  180
mBEC  60  180
mBEC  120
1
mBEC  mAD  mBC 
2
1
120  140  mBC 
2
240  140  mBC
mBC  100
2.)
mAC  360  260
mAC  100
1
mB  mAEC  mAC 
2
1
mB  260  100
2
1
mB  160
2
mB  80
3.)
5.)
mADC  360  140
mADC  220
1
mB  mADC  mAC 
2
1
mB  220  140
2
1
mB  80
2
mB  40
m1  20
m2  45
m3  65
m4  20
AF  40
DE  10
4.)
6.)
mAD  360  150  50
mAD  160
1
mB  mAD  mAC 
2
1
mB  160  50
2
1
mB  110
2
mB  55
m1  40
m2  50
m3  60
m4  20
m5  85
m6  30