Page 40 - Ganit Kaushal

5. How many lines can be drawn through a given point?
6. How many lines can be drawn through two distinct given points?

7. How many lines can be drawn through three collinear points? A
8. In the adjoining figure,
(a) Which line segments intersect at D? E
(b) Which line segments intersect at B? B F
(c) What other line segments can be drawn?
(d) Name the point of intersection of line segments AC and BE.
9. Draw a rough figure and label it suitably in each of the following cases: C D

(a) Point P lies on AB. (b) XY and PQ intersect at M.

(c) Line l contains E and F but not D. (d) OP and OQ meet at O.
2.3 ANGLES
An angle is formed by two rays having a common initial point. In the adjacent figure, B

there is an angle formed by rays OA and OB, where O is the common starting point. arm

The common point O is called the vertex of the angle, and the rays OA and OB are
generally known as the arms of the angle.
O arm A
Sometimes, arms are also referred to as the legs of the angle. Since the angle is made (vertex)
at the vertex O, it can simply be called angle O.
But it can be confusing if more than one angle is made at a vertex (as shown in the C
adjacent figure). Here, angle O will not clarify whether we are talking about angle AOB arm B
or angle BOC or angle AOC. So, to be more precise, we write it as angle AOB. The arm
symbol ‘∠’ is used to replace the word angle, and we write ∠AOB. The vertex name
is always written in the middle. ∠AOB and ∠BOA represent the same angle. As shown
in both figures, a small curve is used to indicate the angle. O arm A
An angle can also be assumed as made by rotating the initial ray about its vertex in an anticlockwise sense,
i.e., the sense in the direction opposite to the movement of the hands of a clock. It is shown by the arrowed
arc in the figure.
The sense of rotation in the direction of the movement of the hands of a clock is Final position of ray
known as a clockwise sense of rotation. The amount of rotation decides the size
of the angle. The more the rotation, the larger is the size of the angle. The size arm Amount of turn is
of the angle is measured as the amount of rotation between the initial position the size of the angle
of the ray (initial ray) and the final position of the ray (final ray) as shown in Vertex Initial position of ray
the adjoining figure.

Is it possible to divide a given angle into two equal parts?
You got it right. The answer is yes.
It can be done with the help of a ray called the bisecting ray or the bisector of the given Final ray A
angle. It is drawn between the initial and final ray with the same vertex such that the
two newly formed angles are equal. α Bisector C
O α
In the adjoining figure, the bisector ray OC is drawn between rays OA and OB such that
∠AOC = ∠BOC = α. Initial ray B

This process of getting half of a given angle is called bisecting the angle.

38 Mathematics-6