CBSE Notes Class 7 Maths Chapter 5 Lines and Angles

This chapter introduces the basic concepts of lines and angles, which form the foundation of geometry. Understanding how lines and angles interact helps in solving problems related to shapes, patterns, and measurements. It covers the types of lines (such as parallel and intersecting) and angles (like acute, right, and obtuse), along with their properties and relationships. Through this chapter, students will gain insights into fundamental geometric concepts that are applicable in both theoretical and practical scenarios.

1.0Line

A line is defined as a straight path consisting of points that extends infinitely in both directions. It possesses only one dimension, meaning its length cannot be measured. To illustrate that a line continues endlessly in both directions, arrowheads are placed at each end. A line is identified using any two points located on it; for example, it can be referred to as line AB or line BA, denoted as $\overleftrightarrow{\mathrm{AB}}$ or $\overleftrightarrow{\mathrm{BA}}$. For any two distinct points, there exists precisely one line that passes through them. 

2.0Line Segment

The line segment AB, denoted as $\overline{AB}$, consists of the endpoints A and B, along with all the points located between them. It's important to note that this segment can also be referred to as $\overline{BA}$ . 

3.0Ray

The ray AB, represented as AB, includes the endpoint A and all points on the ray that are on the same side of A as point B. It's important to note that rays AB and BA are distinct from one another. 

If point C is located on ray AB between A and B, then rays CA and CB are considered opposite rays. 

Segments and rays are classified as collinear if they exist on the same line, which also means that opposite rays are collinear. Furthermore, lines, segments, and rays are coplanar if they all lie within the same plane.

4.0Angles

An angle is an inclination between two rays with the same initial point. The initial point is the vertex and the two rays are the arms of the angle. An angle is represented by the symbol ∠.

1 rotation = 360° 

1° = 60’ = 60 minutes 

1’ = 60” = 60 seconds

Types of Angles

1. Acute Angle: 

An angle whose magnitude is more than 0° but less than 90° is called an acute angle.

2. Right Angle: 

An angle whose magnitude is 90° is called a right angle.

3. Obtuse Angle:

An angle whose magnitude is more than 90° but less than 180° is called an obtuse angle.

4. Straight Angle:

An angle whose magnitude is 180° is called a straight angle.

5. Reflex Angle: 

An angle whose magnitude lies between 180° and 360° is called a reflex angle.

Pair of Angles

1. Adjacent Angles 

Two angles in the same plane are called adjacent if they have a common vertex and a shared side, with no overlap in their interiors.

In the figure, point O serves as the common vertex for angles $\angle AOB$ and $\angle BOC$ , with ray OB acting as their shared side. Since the interiors of these angles do not intersect, $\angle AOB$ and $\angle BOC$ are classified as adjacent angles.

2. Linear pair 

Two adjacent angles whose sum is 180°, forms a linear pair. 

AOB is a straight line. 

⇒ ∠AOB = 180° 

⇒ ∠AOC + ∠BOC = 180° 

Hence, ∠AOC and ∠BOC makes a linear pair.

3. Vertically Opposite Angles 

When two lines intersect at a point, four angles are formed. Angle 1 is opposite to angle 3, angle 2 is opposite to angle 4. 

Pairs of angles such as angles 1 and 3, and angles 2 and 4 are called vertically opposite angles or simply vertical angles. 

If two lines intersect then vertically opposite angles are equal.

4. Supplementary and Complementary Angles:

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

∴ ∠AOC + ∠BOC = 180°

If the sum of two angles is 90°, they are called complementary angles. 

∴ ∠AOC + ∠BOC = 90°

5. Sum of the angles around a point:

The sum of all the angles at a point each being adjacent to the next is 360° or 4 right angles. In figure, ∠AOB + ∠BOC + ∠COD + ∠DOA = 360°

5.0Pair of Lines

1. Intersecting Lines: 

Two or more lines that meet at a common point are called intersecting lines, and the point where they meet is known as the point of intersection. They intersect at exactly one point, regardless of the angle. 

If lines never meet at any point, they are called non-intersecting lines.

2. Parallel Lines

Two lines in the same plane are called parallel if they never intersect, no matter how far they are extended.  

If line $\ell$ is parallel to line m, it is written as $\ell \Vert m$ . The distance between parallel lines remains constant and equals the perpendicular distance between them.

3. Transversal Lines

A transversal is a line that cuts across (intersects) two or more lines in distinct points.

Properties of parallel lines cut by a transversal 

In figure, parallel lines λ and m have been intersected by a transversal x. 

The measurement of the angles using a protractor proves some important properties.