Parallel Lines: Definition, Pair of Angles, Examples

The two lines that do not meet or intersect at any point and always stay at a distance are called parallel lines. We can also say that parallel lines are the same distance apart without meeting each other. For example, when a line intersects through the parallel lines, there is a formation of angles such as corresponding, alternate interior, alternate exterior and interior angles of the same side of the transversal.

Parallel lines are the most important geometrical concepts in mathematics. Students learn the concept in their primary classes in order to solve mathematical problems quickly. You can check NCERT Solutions for Class 7 Maths Chapter 5 for better understanding. We have provided detailed information on parallel lines in this article. Read on to find out about its definition, properties, and examples.

What are Parallel Lines and Pairs of Angles?

Parallel lines are the two lines that do not meet at a point in a plane surface. The two lines intersecting each other at a point are called intersecting lines. And these lines meeting at 90 degrees are called perpendicular lines. Lines that do not cross each other or intersect at a point are parallel lines. It is denoted as ||’.

Pair of Angles

The lines that intersect two or more lines at a distant point are called a transversal. For example,

l is a transversal line intersecting m and n at a point P and Q. You can see there is the formation of four angles at point P and Q. ∠ 1, ∠ 2, ∠ 7 and ∠ 8 are exterior angles whereas, ∠ 3, ∠ 4, ∠ 5 and ∠ 6 are interior angles. Some of the angles formed when the transversal line intersects the two lines are as follows:

1. Corresponding angles: (i) ∠ 1 and ∠ 5 (ii) ∠ 2 and ∠ 6 (iii) ∠ 4 and ∠ 8 (iv) ∠ 3 and ∠ 7
2. Alternate interior angles: (i) ∠ 4 and ∠ 6 (ii) ∠ 3 and ∠ 5
3. Alternate exterior angles: (i) ∠ 1 and ∠ 7 (ii) ∠ 2 and ∠ 8
4. Interior angles on the same side of the transversal: (i) ∠ 4 and ∠ 5 (ii) ∠ 3 and ∠ 6

Properties of Parallel Lines

Some of the properties of parallel lines are mentioned below:

1. Corresponding angles of the parallel lines are equal.
2. Alternate interior angles of the parallel lines are equal
3. Vertically opposite angles of the parallel lines are equal.
4. Lines that are parallel at a given point are parallel to each other.

Theorems of Parallel Lines

The theorems of parallel lines are as follows:

1. If two lines intersect each other, then the vertically opposite angles are equal.
2. If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.
3. If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.
4. If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.
5. If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.

Parallel Lines: Solved Examples

Some of the solved examples of parallel lines are given below:

Example 1: In the given figures below, decide whether ? is parallel to ?.

Solution: ? + 123 ° = 180 ° (Linear Pair) ⇒ ? = 180 ° − 123 ° = 57 °

For the lines ? and ? to be parallel to each other, their corresponding angles ∠ABC and ? should be equal. Since they are equal.

Therefore, ? is parallel to ?.

Example 2: In the adjoining figure, identify

(i) the pairs of corresponding angles.

(ii) the pairs of alternate interior angles.

(iii) the pairs of interior angles on the same side of the transversal.

(iv) the vertically opposite angles.

Solution:

(i) the pairs of corresponding angles: ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8

(ii) the pairs of alternate interior angles: ∠2 and ∠8, ∠3 and ∠5

(iii) the pairs of interior angles on the same side of the transversal: ∠2 and ∠5, ∠3 and ∠8

(iv) the vertically opposite angles: ∠1 and ∠3, ∠2 and ∠4, ∠5 and ∠7, ∠6 and ∠8

Also, Check:

Summary

Parallel lines are defined as the two lines that do not meet at a point in a plane surface. It is symbolised as ‘||’. Transversal is defined as the lines that intersect two or more lines at a distant point. It is important to note that if two lines intersect each other, then the vertically opposite angles are equal. If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal. Corresponding angles, alternate interior angles, and vertically opposite angles of the parallel lines are equal.

FAQs on Parallel Lines

Q.1. What are parallel lines?
A. The two lines that do not meet or intersect at any point and always stay at a distance is called parallel lines.

Q.2. What are the types of pair of angles in parallel lines?
A. The pair of angles of in parallel lines are corresponding, alternate interior, alternate exterior and interior angles of the same side of the transversal.

Q.3. What are the properties of parallel lines?
A. The properties of parallel lines are as follows:
1. Corresponding angles of the parallel lines are equal.
2. Alternate interior angles of the parallel lines are equal
3. Vertically opposite angles of the parallel lines are equal.
4. Lines that are parallel at a given point are parallel to each other.

Q.4. What is the symbol of parallel lines?
A. The symbol of a parallel line is ‘||’.

Q.5. What are the types of angles in parallel lines?
A. The types of angles in parallel lines are corresponding angles, alternate interior angles, alternate exterior angles, and interior angles on the same side of the transversal.

We have now provided you with detailed information on parallel lines in this article. Make the best use of these solutions to solve the exercise problems. Also, do the NCERT textbook activities to have a hands-on experience of the processes you read. A proper understanding of the concepts will also lay a good foundation for your future grades.

We hope this article on Parallel Lines helps you. If you have any questions feel to post it in the comment below. We will get back to you at the earliest.