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8/2/2019 Properties of Angles in Circles
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PROPERTIES OFANGLES IN CIRCLES
8/2/2019 Properties of Angles in Circles
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Angles At Centre and Circumference
An angle at the centre ofa circle is an angle withits vertex at the centreand two radii as its
sides.
An angle at thecircumference is an
angle with its vertex atthe circumference of thecircle and two chords asits sides.
Q
P
C
A
AQC
APC
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Example:
Name the angles that are subtended by arc PQ at
the centre and at the circumference respectively.a) b)
QP
y
x
O
g
f
e
S
R
Q
P
O
Angles at the centre
= ________
Angle at the circumference
= ________
Angles at the centre
= ________
Angle at the circumference
= ________
y
x
e
f
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Angles Subtended By Same Arc
Angles subtended atthe circumference bythe same arc are
equal.z
y
x
Q
P
x = y = z because
these angles aresubtended by the same
arc PQ (a.k.aangles in
the same segment )
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Example:
Name the angles that are of the same size.
a) b)
O
zu
y
x
w
zx
y
O
x = y y = z
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Angles Subtended By Arcs
Of Same Length
The angles subtended at the circumference by arcs of
the same length are equal.
R
P
Q
yx
B
C
A
When length of minor arc AC = length
of minor arc PR, then x = y.
When x = y, then length of minor arc AC =length of minor arc PR.
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The angles subtended at the centre by arcs of the samelength are equal.
O
y
x
Q
P
B
A
When length of minor arc ab =
length of minor arc PQ, then x = y.
When x = y, then length of minor
arc AB = length of minor arc PQ.
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Example:
Name the angles that are of the same size in each
of the following.a) b)
xy
O
u
z w
v
u
O
x = y v = w
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Relationship Between Angles
Angle subtended by an arc at its centre, istwice the angle at the circumference.
Angle at the centre
= 2 x Angle at the circumference
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Example:
O
2x
x
O
2x
x
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Example:
2x
x
x
OO
2x
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Example:
Find the value of x.
a)
x
35O
x = 2 x 35
x = 70
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b)
xO
120
x = 2 x 120
x = 240
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c)
x60
O
x =60
2
x = 30
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d)
x
280
O
x =280
2
x = 140
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Angles In A Semicircle
Any angle at the circumference of a semicircle is aright angle (90).
Example:
Q
P
BAO
Since AOB is the diameter;then APB =AQB =90.
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Example:
Find the value of x.
a)
x38
x = 180 - 90 - 38x = 58
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b)
xx =
180 - 90
2x = 45
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c)
x = 90 - 42
x = 48
42
x
O
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Solving ProblemsExample:
1)
C
B
A
110
X
O
In the figure, O is thecentre of the circle.
Find the value of x.Reflex angle AOC= 2 x 110
= 220x = 360 - 220x = 140
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2)
O
60
x
D
B
CA
In the figure, O is thecentre of the circle.Find the value of x.
ABC = 90CBD = 90 - 60
= 30
x = 30(Angles in thesame segment)
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3)
T
D
C
B
A30
x
O
In the figure, O is the centreof the circle. ATC and BOD arestraight lines. Find the valueof x.
BOC = 2 x 50= 60
x = 180 - 60= 120
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4)T
S
R Q P
x
110
50
O
In the figure, SOQ is thediameter of the circle
with centre O. Find thevalue of x.
RSQ = 50
QRS = 90
RQS = 180 - 90 - 50= 40
x = 180 -110 - 40= 30