**“Math Class Study Notes for Students – Angle in a Circle,”**

31 Angles in a Circle

By studying this lesson you will be able to
identify and apply the theorems related to angles in a circle.


31.1 Angles subtended by an arc at the centre and on the circle

Major arc


O



A B
Minor arc


The circle in the figure is divided into two parts by the two points A and B on the
circle. These two parts are called arcs. When the two points A and B are such that
the straight line joining the two points passes through the centre of the circle, that
is, when it is a diameter, then the two arcs are equal in length. When this is not the
case, the two arcs are unequal in length. Then the shorter arc is called the minor arc
and the longer arc is called the major arc.

The angle which is formed by joining the end points of
O the minor arc AB which is indicated by the thick line in the
figure, to the centre, is defined as the angle subtended by the
minor arc AB at the centre.
A B




The reflex angle which is formed by joining the end points
of the major arc AB which is indicated by the thick line in the
O figure, to the centre, is defined as the angle subtended by the
major arc AB at the centre.
A B

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,Note: The angle subtended at the centre by a major arc is always a reflex angle.

C
Let us assume that C is any point on the major arc AB.
The angle ACB ˆ is formed when the end points of the minor arc
AB are joined to the point C on the major arc. That is, ACBˆ is
the angle that is subtended by the minor arc AB on the remaining
A B
part of the circle.



Similarly, the angle ADBˆ in the given figure is defined as the
angle that is subtended on the remaining part of the circle by the
major arc AB.
A B
D


Example 1
C



O

R
P
Q
The centre of the circle in the given figure is O.
(a) Write down,
^i& the angle that is subtended by the minor arc PR on the remaining part of the
circle.
^ii& the angle that is subtended at the centre by the minor arc PR.
(b) Write down,
^i& the angle that is subtended by the major arc PR on the remaining part of the
circle.
^ii& the angle that is subtended at the centre by the major arc PR.
^a& ^i& P
^ii&
ˆ
^b& ^i& PQR
^ii& reflex angle
For free distribution 165

, Exercise 31.1
1. Copy the four circles in the figure given below onto your exercise book. The
centre of each circle is denoted by O.


O O O O


A B A B A B A B

a b c d
Mark each of the angles indicated below.
(i) An angle that is subtended on the remaining part of the circle by the minor arc
in figure a.
(ii) The angle subtended at the centre by the minor arc in figure b.
(iii) An angle that is subtended on the remaining part of the circle by the major arc
in figure c.
(iv) The angle subtended at the centre by the major arc in figure d.

A
2. According to the figure,
(i) write down for the minor arc AB,
X O
(a) the angle subtended on the remaining part of the circle
Y
(b) the angle subtended at the centre.
B
(ii) write down for the major arc AB,
(a) the angle subtended on the remaining part of the circle
(b) the angle subtended at the centre.

3. The centre of the circle in the given figure is O. The point X is on the major arc
PQ.

^i& Write down the angle that is subtended on the remaining
part of the circle by the minor arc PQ.
O
^ii& Write down the angle that is subtended at the centre by the
Q minor arc PQ.
X

P
^iii& Mark any point on the minor arc PQ and name it Y. Define
<




the angle PYQ .
^iv& Write down the angle that is subtended at the centre of the
circle by the major arc PQ.
166 For free distribution