NCERT Book Class 6 Maths Lines and Angles PDF Download

Chapter 2: Lines and Angles

In this chapter, we will explore some of the most basic ideas of geometry including points, lines, rays, line segments and angles. These ideas form the building blocks of 'plane geometry', and will help us in understanding more advanced topics in geometry such as the construction and analysis of different shapes.

2.1 Point

Mark a dot on the paper with a sharp tip of a pencil. The sharper the tip, the thinner will be the dot. This tiny dot will give you an idea of a point. A point determines a precise location, but it has no length, breadth or height. Some models for a point are given below.

[Figure: Models for a point - the tip of a compass, the sharpened end of a pencil, the pointed end of a needle, See in your textbook]

If you mark three points on a piece of paper, you may be required to distinguish these three points. For this purpose, each of the three points may be denoted by a single capital letter such as Z, P and T. These points are read as 'Point Z', 'Point P' and 'Point T'. Of course, the dots represent precise locations and must be imagined to be invisibly thin.

Teacher's Note

A point has no size, no length, and no width - it is just a location. When you draw a dot with a pencil, you are drawing a picture of where a point is, but the actual mathematical point is infinitely small. Remember this when you label points in diagrams.

2.2 Line Segment

Fold a piece of paper and unfold it. Do you see a crease? This gives the idea of a line segment. It has two end points, A and B.

Mark any two points A and B on a sheet of paper. Try to connect A to B by various routes (Fig. 2.1).

What is the shortest route from A to B? This shortest path from point A to Point B (including A and B) as shown here is called the line segment from A to B. It is denoted by either AB or BA. The points A and B are called the end points of the line segment AB.

[Figure 2.1: Various routes connecting points A and B, showing the shortest direct path, See in your textbook]

2.3 Line

Imagine that the line segment from A to B (i.e., AB) is extended beyond A in one direction and beyond B in the other direction without any end (see Fig. 2.2). This is a model for a line. Do you think you can draw a complete picture of a line? No. Why?

A line through two points A and B is written as \( \overleftrightarrow{AB} \). It extends forever in both directions. Sometimes a line is denoted by a letter like \( l \) or \( m \).

Observe that any two points determine a unique line that passes through both of them.

[Figure 2.2: A line extending infinitely in both directions through points A and B, labeled m, See in your textbook]

Teacher's Note

Notice the difference: a line segment AB has two end points and you can measure its length, but a line \( \overleftrightarrow{AB} \) goes on forever in both directions and has no endpoints. If you see arrows on both ends of a line in a diagram, that means it is a line, not a line segment.

2.4 Ray

A ray is a portion of a line that starts at one point (called the starting point or initial point of the ray) and goes on endlessly in a direction. The following are some models for a ray:

[Figure: Models for a ray - beam of light from a lighthouse, ray of light from a torch, sun rays, See in your textbook]

Look at the diagram (Fig. 2.3) of a ray. Two points are marked on it. One is the starting point A and the other is a point P on the path of the ray. We then denote the ray by \( \overrightarrow{AP} \).

[Figure 2.3: A ray starting at point A and passing through point P, See in your textbook]

Figure it Out

1. Rihan marked a point on a piece of paper. How many lines can he draw that pass through the point?

   Sheetal marked two points on a piece of paper. How many different lines can she draw that pass through both of the points?

   Can you help Rihan and Sheetal find their answers?

Key Points

• A point marks a precise location but has no length, breadth, or height.
• A line segment is the shortest path between two points and has two end points; a line extends forever in both directions through any two points.
• A ray starts at one point and extends endlessly in one direction, and is named by its starting point and another point on its path.

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