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Angles in CirclesAngles in Circles
Objectives:B Grade Use the angle properties of a circle.
A Grade Prove the angle properties of a circle.
Angles in CirclesAngles in Circles
Parts of a circle
Radius
diameter
circumference
Major Sector
Major arc
Minor arc
Minor segment
Major segment
Minor Sector
A line drawn at right angles tothe radius at the circumference is
called theTangent
Angles in CirclesAngles in Circles
Key words:
Subtend: an angle subtended by an arc is one whose two rays pass through the end points of the arc
arctwo rays
angle subtended by the arc
end points of the arc
arcend points of the arctwo rays
angle subtended
Supplementary: two angles are supplementary if they add upto 180o
Cyclic Quadrilateral: a quadrilateral whose 4 vertices lie on thecircumference of a circle
Angles in CirclesAngles in Circles
The angle subtended in a semicircle is a right angle
Theorem 1:
Angles in CirclesAngles in CirclesNow do these:
a70o b45o
c 130ox 2x
a = 180-(90+70)a = 20o
b = 180-(90+45)b = 45o
180-130 = 50o
c = 180-(90+50)c = 40o
3x = 180-90 x = 30o
Angles in CirclesAngles in Circles
The angle subtended by an arc at the centre of a circle is twice thatat the circumference
Theorem 2:
This can also appear like
a
2a
arc
a2a
arc
2a
a
or
Angles in CirclesAngles in Circles
d = 80 ÷ 2d = 40o
e = 72 × 2e = 144o
f = 78 ÷ 2f = 39o
67 ÷ 2 = 33.5o
g = 180-33.5g = 147.5
Now do these:
d
80o
f78o
72o
e
g67o
96o
h
h = 96 × 2h = 192o
Angles in CirclesAngles in Circles
The opposite angles in a cyclic quadrilateral are supplementary(add up to 180o)
Theorem 3:
a
b
c
d
a + d = 180o
b + c = 180o
Angles in CirclesAngles in Circles
i = 180 – 83 = 97o
j = 180 – 115 = 65o
k = 180 – 123 = 57o
l = m = 180 ÷ 2 = 90o
115o
i
j
Now do these:
83o
123o
k
l m
Because the quadrilateral is a kite
Angles in CirclesAngles in Circles
Angles subtended by the same arc (or chord) are equalTheorem 4:
same arcsame angle
same arc same angle
n = 15o
p = 43o
q = 37o
r = 54o
s = 180 – (37 + 54) = 89o
Angles in CirclesAngles in CirclesNow do these:
15o
p
43o
n
r
54o
q
37o
s
Angles in CirclesAngles in CirclesSummary
The angle subtended in a semicircle is a right angle The angle subtended by an arc at the centre of a circle is twice that at the circumference
a
2a
arcThe opposite angles in a cyclic quadrilateral are Supplementary (add up to 180o)
a
b
c
d
a + d = 180o
b + c = 180o
Angles subtended by the same arc (or chord) are equal
More complex problemsAngles in CirclesAngles in Circles
64o
a
b
c
d
e65o
f28o
a = 90o
The angle subtended in a semicircle is a right angle
Angle AED is supplementary to angle ACDb = 180 – 64 = 116o
A
B
C
D
E
O
Cyclic quadrilateral ACDE
Angle ABD is supplementary to angle AEDb = 180 – 116 = 64o
Cyclic quadrilateral ABDE
The angle subtended by an arc at the centre of a circle is twice that at the circumference
d = 130o
The angle subtended by an arc at the centre of a circle is twice that at the circumference
orAngles subtended by the same arc (or chord) are equal
e = 65o
Opposite angles are equal, therefore triangles FGX andHXI are congruent.
F
GH
I
X
f = 28o
a70o b45oc 130o
x 2x
Worksheet 1Angles in CirclesAngles in Circles
a = b = c = x =
d
80o
f78o
72o
e g67o
96o
h
d = e = f = g =
h =
Worksheet 2Angles in CirclesAngles in Circles
115o
i
j
83o
123o
k
l m
15o
p
43o
n
r
54o
q
37o
s
i =
j = k=
l =
m =
n =
p =
q =
r =
s =
Worksheet 3Angles in CirclesAngles in Circles
64o
a
b
c
d
e65o
f28o
A
B
C
D
E
O
F
H
I
a =
b =
c =
d =
e =
f =