# Properties of Parallel Lines

Before talking about lines that are parallel to the same line, let us recall what parallel lines are. Non-intersecting or parallel lines are the lines that do not intersect each other. They are always at the same distance from one another. Hence, they never meet. Say you are given a line A that is parallel to a line. There is another line B which is parallel to the same line. Does this imply that lines A and B are parallel to each other? We will discuss it in this article.

## Lines Parallel to the Same Line

To check whether lines parallel to the same line are parallel or not, let us take an example. In the following figure, we are given that line a and line c are parallel to line b.

Since a || b, so âˆ 1 = âˆ 2 Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (Corresponding angles axiom)

Since c || b, so âˆ 3 = âˆ 2Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â  (Corresponding angles axiom)

Therefore, âˆ 1 = âˆ 3 Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â (Commutative property)

But âˆ 1 and âˆ 3 are corresponding angles and they are equal.

So by the converse of corresponding angles axiom, it can be deduced that a || c.

### Parallel Lines: Theorem

• The lines which are parallel to the same line are parallel to each other as well.
• This property holds good for more than 2 lines also.

Example

In the following figure, m, n,Â and l are parallel lines. And AB is parallel to CD. Find the value of angle x using the given angles.

Solution:

Since m || l, âˆ CGI = 120Â° Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â (Corresponding angles)

âˆ FGC = 180Â° – 120Â° = 60Â° Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (Linear pair with âˆ CGI)

âˆ BIH = âˆ BFE = 40Â° Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  (Corresponding angles)

âˆ BFE = âˆ GFC = 40Â° Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â  Â (Vertically opposite angles)

Therefore, x + 40Â° + 60Â° = 180Â° Â  Â  Â  Â  Â  Â  Â  Â  Â  Â (Sum of interior angles of a triangle is 180Â°)

Or x = 180Â° – 100Â° = 80Â°

### Conditions for Lines to be parallel

If two straight lines are cut by a transversal,

• the pair of alternate angles are equal, then two straight lines are parallel to each other.
• the pair of interior angles on the same side of traversals is supplementary, then the two straight lines are parallel.
• the pair of corresponding angles are equal, then the two straight lines are parallel to each other.

## Frequently Asked Questions â€“ FAQs

Q1

### When do we say two lines are parallel to each other?

Two lines are said to be parallel if they donâ€™t intersect with each other
Q2

### If corresponding angles are equal then lines are?

When corresponding angles are equal then lines are parallel.
Q3

### What is the sum of interior angles in a triangle?

The sum of interior angles in a triangle is 180 degrees.
Q4

### What is the measure of other angle, in a straight line when one angle is 54 degrees?

The sum of two angles in a straight line is 180 degrees
If one angle is 54 degrees then another angle will be 180-54=126 degrees.
Q5

### If the pair of corresponding angles are equal then lines are parallel. True or False?

True. If the pair of corresponding angles are equal then lines are parallel.

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