Grade IX - Mathematics

Author:
Rahul Chandra Saha
M.Sc., Mathematics

Introduction and basic terms​

Introduction

Everything we see around us is 3 dimensional. The chair you are sitting on, the mobile or laptop you are using, the trees all are 3 dimensional. That is each of the objects has length, breadth and height.

Now what if we reduce the dimension to two? Have you seen something like that? With only two dimensions? Yes, it’s the wall of your room, the pictures – all are two dimensional,called planes.

Now what if we further reduce the dimension to one? You can see it also.
It’s the intersection of the walls of your room, called the lines or line segments.

Think Yourself : What if the dimension becomes zero?

You have an idea about the lines. By definition a straight line or simply a line is the set of all points between and extending between two points. A minimum of two points is required to draw a line. We have read that light moves in a straight line or straight line is the shortest distance between any two points. So concept of lines are an important tool not only to build mathematical concepts but also to discuss various physical phenomenons.
In this chapter we will discuss about lines and intersecting lines.

Basic terms and definitions

Line segment

A line with two end points.

Ray

A line with one end point.

Notation

A line is denoted  by $\overleftrightarrow{AB}$. A ray is denoted by $\overrightarrow{AB}$. A line segment is denoted by $\overline{AB}$.

Collinear and Non-collinear points

The points lying in the same straight line are called collinear points. Thus three or more points are collinear if we can draw a line choosing any two of the points and the third lies on the line drawn. If not the the lines are  non collinear points.
Note. Two points will always be collinear.

Angle

When two rays evolve from same end point an angle is formed. Here $\overrightarrow{EF}$ and   $\overrightarrow{EG}$ are two rays with E as the same end point(or source) and $\angle{FEG}$ is formed. The end point E is called the vertex and $\overrightarrow{EF}$ and  $\overrightarrow{EG}$ are the arms.

Unit of an Angle 

The most contemporary units of angle are degree(°)and radian(rad).

Types of Angles. 

There are different types of angles based on measure and position to each other of the angles –

Acute Angle
If the measure of an angle is in between 0° and 90°, the angle is an acute angle.
E.g. 45°, 87° are acute angles.

Right Angle
If the measure of an angle exactly 90° the angle is called a right angle.

Obtuse Angle
If the measure of an angle is in between 90° and 180°, the angle is an acute
angle.
E.g. 145°, 177° are obtuse angles.

Complementary Angles

If the sum of two angles is 90° then the angles are called complementary to each other.
E.g. angles 55° and 35° are complementary to each other. 

Supplementary Angles 

If the sum of two angles is 180° then the angles are called supplementary angles.

Adjacent Angles 

If the source or vertex of two angles are same and they have a common arm, the angles are called adjacent angle. $\angle{BAC}$  and  $\angle{CAD}$ are adjacent angles.

Linear pair of Angles

Two adjacent angles are called linear pair of angles if their sum is 180°.

$\angle{AEC,} \angle{CEB}$ are linear pair of angles.

Vertically opposite Angles
When two lines intersect each other the angles opposite to each other are called vertically opposite angles.

$\angle{AEC}$ and $\angle{BED}$ are vertically opposite angles.