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