An Introduction

Lines are said to be parallel when they do not intersect each other at any point. There are various methods to check whether the lines are parallel or not. We can check it simply by the construction method. We can extend the lines in the same direction if the lines do not intersect each other they will simply be called parallel lines.

Definition of Parallel Lines

Lines are called parallel if they never intersect each other at any point. Lines are either parallel or intersecting. Parallel lines extended in the same direction never intersect each other. We can see in the image that line AB is parallel to CD.

Line AB is parallel to line CD

Types of angles in parallel lines

There are the following types of angles in parallel lines: 

• Corresponding angles

• Alternate interior angles

• Alternate exterior angles

• Consecutive interior angles

The different types of angles made by a transversal

Corresponding angles: Corresponding angles are the angles, which hold the same correlative position simultaneously to the transversal line. In figure 2, angles 1 and 2, 3 and 4, 5 and 6 , and 7 and 8 are corresponding angles.

Alternate interior angles: Alternate interior angles are formed when a transversal line passes through two parallel lines. They are in the interior part of figure 2. In the figure 2, the angles 2 and 7, and 4 and 5 are alternate interior angles.

Alternate exterior angles: Alternate exterior angles are formed in the exterior part of the lines when a transversal line passes through two parallel lines. In figure 2, the angles1 and 8 and 3 and 6are alternate exterior angles

Consecutive interior angles: Consecutive interior angles are those angles that are consecutive to each other and are on the same side of the transversal. They are complementary to each other. In figure 2, the angles are 5 and 2 and 4 and 7 consecutive interior angles.  

How to prove two lines are parallel

There are different methods to prove that two lines are parallel, we shall use the corresponding angle axiom to prove that the two lines are parallel. In this theorem when a transversal intersects two lines and alternate angles are congruent, the lines are parallel to each other.  

We have given that in the figure, a transversal    intersects two parallel lines and their alternate interior angles are equal,  

                       1=3 

                       1=3 (vertically opposite angles)

Therefore, from the above equations, we can write

                      2 = 3

Here that angles 2 and 3 are corresponding,

 By the application of the corresponding angle axiom, 

  AB is parallel to CD. 

The alternate and corresponding angles made by the transversal

Pair of parallel lines

A pair of parallel lines is the relation between two lines that have the same slope or we can say that they will never intersect. There may be any number of lines in a figure that can be parallel. If a transversal intersects the pair of lines and the corresponding angles, alternate angles and the consecutive interior angles formed are equal, the lines are parallel to each other.

Parallel lines examples

We all have seen parallel lines in books and we can draw them in our notebooks. In our surroundings, we can see many examples, some of the examples of parallel lines are:

• Railway tracks 

• Zebra crossing on the road

• Lines on the notebook 

• Cricket stumps

• Opposite boundaries of an eraser and box

• Opposite edges of the ruler

Interesting Facts

• The parallel lines are equal in distance from each other, or we can say they are equidistant from each other.

• They can be extended up to an indefinite distance.

• The parallel lines never meet each other at any point.

• The parallel lines are coplanar.

Key Features

• Two lines are parallel when they do not intersect each other at any point.

• When a transversal intersects two parallel lines, corresponding angles, alternate angles, and the consecutive interior angles formed are equal.

• We can show the two lines parallel by the converse theorem of the interior, exterior, corresponding, and consecutive angles.

Solved Questions 

Q1. Line AB is parallel to CD, and PQ is parallel to SR and APQ= 500. What are the values of angles x, y, and z?

Ans: x = 500 (corresponding angles are equal)

z = x (corresponding angles are equal)

z =  500

z + y =  1800  (sum of the angles of complementary angles equals 1800)

 500 + y =  1800

y =  1800- 500

y = 1300

Q2. In the figure, we have lines AB and CD parallel, PS is the transversal. Find DRS.

Ans: ∠Q + 450 = 1800(sum of the angle is 180 degrees)

∠Q  = 1800 -  450

∠Q  = 1350

Q3. Two supplementary angles are in the ratio 3:2. Find the value of both angles.

Ans: The sum of the complementary angles is 1800.

Let the first angle be 3x and the second angle 2x.

Then, 3x + 2x = 1800

5x = 1800

x = 360

First angle is 3 x 360 = 1080

Second Angle is 2 x 360 = 720

Q3. Two supplementary angles are in the ratio 3:2. Find the value of both angles.

Ans: The sum of the complementary angles is 1800.

Let the first angle be 3x and the second angle 2x.

Then, 3x + 2x = 1800

5x = 1800

x = 360

First angle is 3 x 360 = 1080

Second Angle is 2 x 360 = 720

Frequently Asked Questions

Q1. Name the mathematician who talked about parallel lines.

Ans: Euclid gave the study of pairs of parallel lines. 

Q2. Why do the pair of parallel lines never intersect?

Ans: The parallel lines do not intersect with each other because they have the same slope. 

Q3. What is the purpose of studying parallel lines?

Ans: We study the concept of parallel lines to understand the path of the objects and the different shapes. It is widely applicable in construction and engineering.

Q4. What is the angle between two parallel lines?

Ans: The angle between two parallel lines is Zero degrees or 180 degrees.