Properties of Parallel Lines

What are Parallel Lines?

Two lines in a plane are said to be parallel if they do not intersect, when extended infinitely in both the direction.

Also, the distance between the two lines is the same throughout.

$Parallel Line$

Parallel Lines

The symbol for denoting parallel lines is ∥.

If lines l and m are parallel to each other, we can write it as l∥m and which is read as ‘l is parallel to m’.

Properties Of Angles Associated with Parallel Lines:

$angles associated with parallel lines$

If two parallel lines are cut by a transversal, then

• the pair of corresponding angles is equal (∠2 = ∠6); (∠3 = ∠7); (∠1 = ∠5); (∠4 = ∠8).

• the pair of interior alternate angles is equal (∠4 = ∠6); (∠3 = ∠5).

• the pair of exterior alternate angles is equal (∠1 = ∠7); (∠2 = ∠8).

• interior angles on the same side of transversal are supplementary, i.e., ∠3 + ∠6 = 180° and ∠4 + ∠5 = 180°.

For example let us observe, the adjacent figure shows two parallel straight lines AB and CD. When two parallel lines AB and CD are cut by a transversal MN.

$two parallel straight lines$

(i) Interior and exterior alternate angles are equal.

i.e. ∠3 = ∠6 and ∠4 = ∠5 [Interior alternate angles]

∠1 = ∠8 and ∠2 = ∠7 [Exterior alternate angles]

(ii) Corresponding angles are equal.

i.e. ∠1 = ∠5; ∠2 = ∠6; ∠3 = ∠7 and ∠4 = ∠8

(iii) Co-interior or allied angles are supplementary.

i.e. ∠3 + ∠5 = 180° and ∠4 + ∠6 = 180°

Conditions of Parallelism:

If two straight lines are cut by a transversal, and if

• the pair of corresponding angles is equal, then the two straight lines are parallel to each other.

• the pair of alternate angles is equal, then the two straight lines are parallel to each other.

• the pair of interior angles on the same side of transversal is supplementary, then the two straight lines are parallel.

Therefore, in order to prove that the given lines are parallel; show either alternate angles are equal or, corresponding angles are equal or, the co-interior angles are supplementary.

Parallel Rays:

Two rays are parallel if the corresponding lines determined by them are parallel. In other words, two rays in the same plane are parallel if they do not intersect each other even if extended indefinitely beyond their initial points.

$parallel rays$

Parallel Rays

Therefore, ray AB ∥ ray MN

Parallel Segments:

Two segments are parallel if the corresponding lines determined by them are parallel.
In other word, two segments which are in the same plane and do not intersect each other even if extended indefinitely in both directions are said to be parallel.

$parallel segments$