Page 43 - Ganit Kaushal

Comparing Angles by Circular Aid

Let us have a circular transparent sheet which can be moved B Y
and placed on figures. Can we use this for the comparison of
angles? The answer is yes, we can. Let us see how.
O O
We will explain it by comparing the above two angles, ∠AOB A X
and ∠XOY. Let us place the circular transparent sheet on
∠AOB in such a way that its centre is on the vertex of ∠AOB. Let us mark the points A and B on the edge of
circle at the points where the arms of ∠AOB pass through the circle as shown in figure (i). Then remove the
circular sheet from ∠AOB as shown in figure (ii).
Now let us place this circular sheet on ∠XOY with the centre at the vertex and arm OX passing through OA,
as shown in figure (iii). Mark the point Y on the edge of the circular sheet.

B B B
Y
R
O O O
A X
A A

( i ) (ii) (iii)
Can you now tell which angle is bigger?

Yes, you got it right, ∠AOB is bigger than ∠XOY, because point B lies after point Y in an anticlockwise
sense.
So, we have easily learnt, how we can use a transparent circular sheet to compare the two given angles.

2.5 INTERIOR, EXTERIOR AND BOUNDARY OF AN ANGLE

Consider an angle, say ∠AOB. The regions associated with this angle can be divided
into three parts as its interior, exterior and boundary. A

The interior of an angle is that part which lies within ∠AOB. In the adjacent figure,
it is shown by the shaded area. It should be noted that the interior is not just a limited Interior
shaded area, it extends indefinitely as the two arms can be extended infinitely.
O B
The exterior of an angle is that part which lies outside ∠AOB. In the adjacent figure,
it is shown by the shaded region.
A
The boundary of ∠AOB is made up of all those points that lie on the arms OA and OB. Exterior

The interior of ∠AOB, together with its boundary, is known as the angular region of
∠AOB.
O B

Lines and Angles 41