If two chords of a circle are equal ,then
their corresponding arcs are congruent and
conversely ,if two arcs are congruent ,
then their corresponding chords are
equal.(in fig 1.1).
D
A
C
B
FIG 1.1
Also the angle subtended by an arc
at the centre is defined to be angle
subtended by the corresponding
chord at the centre in the sense that
the minor arc subtends the angle
and the major arc subtends reflex
angle . Therefore , in fig 1.2 ,the
angle subtended by the major arc
PQ at o is reflex angle POQ .
FIG:1.2
O
P
Q
In view of the property above and theorem the
following result is true Congruent arcs (or equal arcs) of a circle
subtend equal angles at the centre .
Therefore , the angle subtended by a chord of
a circle at its centre is equal to the angles
subtended by the corresponding (minor) arc at
the centre .
The following theorem gives the
relationship between the angles subtended by
an arc at a centre and at a point on
the circle .
THEOREM - The angle subtended by an arc
at the centre is double the angle subtended by it
at any point on the remaining part of the
circle .
Given an arc PQ of a circle
subtending angles POQ at the
O and PAQ at a point a on
the remaining part of the circle
. We need to prove that angle
POQ = 2 angle PAQ .
CASE- (I)
A
O
B
P
FIG:1.3
Q
Case -(II)
A
P
o
B
Q
FIG:1.3
Case – (III)
Q
P
O
B
FIG:1.3
Consider the three different cases as
given in fig 1.3.In (I) , arc PQ is minor
, in (II) , arc PQ is a semicircle and in
(III) , arc PQ is major.
Let us begin by joining AO and
extending it to a point B.
In all the cases ,
^BOQ = ^OAQ + ^AQO
Because an exterior angle of a triangle is
equal to the sum of the two interior
opposite angles.
Also in triangle OAQ ,
OA=OQ (Radii of a circle)
Therefore ,
^OAQ=^OQA
(Theorem)
This gives
^BOQ=2^OAQ
(I)
Similarly ,
^BOP=2^OAP
(II)
From (I) and (II) , ^BOP +^BOQ = 2(^OAP+
^OAQ)
This is same as
(III)
^POQ = 2 ^PAQ
For the case (III) , where PQ is the
major arc , (III) is replaced by reflex
angle POQ = 2 ^PAQ.
C
If ^AOB =110 degree
Then ^ACB =1/2x110
=55 degree.
o
B
A
A
O
B
C
If^AOB= 40 degree
Then ^ACB=1/2x40
=20 degree.
B
A
O
If ^AOB =60 degree
Then,
The reflex angle of
AOB=
=360-60
=300 degree.
If a line segment joining two points
subtends equal angles at two other
points lying on the same side of the line
containing the line segment , the four
points lie on a circle .
OR
The angle subtended by an arc at the
centre is double the angle subtended by
it at any point on the remaining part of
the circle.
1. Find the value of^AXB.(AOB=40degree)
O
X
(ii)
(i)
X
O
A
B
A
B
1.i If ^AOB = 40 degree.
Then ^AXB =1/2 x 40
= 20 .
Therefore ^AXB =20 degree.
ii
If ^AOB = 40 degree
Then ^AXB= 2 x 40
= 80.
Therefore ^AXB = 80 degree
ANSWER IN BRIEF
 How can the angle subtended by an arc at the centre be
defined?
 The angle in semicircle is ______ degree.
 The angle subtended by an arc at the centre is ______
the angle subtended by it at any point on the circle.
 Which is the theorem that defines the relationship
between the angles subtended by an arc at a
centre and at a point on the circle ?
P.S. For answers view the slideshow again