Parallel Lines:

Two lines are said to be parallel when they do not meet at any point in a plane. Lines which do not have a common intersection point and never cross path with each other are parallel to each other. Symbol for showing parallel lines is ||. Two lines which are parallel are represented as \(\overleftrightarrow{AB}\)||\(\overleftrightarrow{CD}\), which means that the line \(\overleftrightarrow{AB}\) is parallel to \(\overleftrightarrow{CD}\). The perpendicular distance between the two parallel lines is always constant.

In the figure shown above, the line segment \(\overline{PQ}\) and \(\overline{RS}\) represent two parallel lines as they have no common intersection point in the given plane. Infinite parallel lines can be drawn parallel to \(\overleftrightarrow{PQ}\) and \(\overleftrightarrow{RS}\) in the given plane.

Lines can either be parallel or intersecting. When two lines meet at a point in a plane, they are known as intersecting lines. If a line intersects two or more lines at distinct points then it is known as a transversal line.

In fig. 2, the line l intersects the lines a and b at points P and Q respectively. The line l is the transversal here.

∠1,∠2,∠7 and ∠8 are the exterior angles and ∠3,∠4,∠5 and ∠6 denote the interior angles.

The angle pairs formed due to intersection by transversal are named as follows:

- Corresponding Angles: ∠1 and ∠6; ∠4 and ∠8; ∠2 and ∠5; ∠3 and ∠7 are the corresponding pair of angles.
- Alternate Interior Angles: ∠4 and ∠5 ; ∠3 and ∠6 denote the pair of alternate interior angles.
- Alternate Exterior Angles: ∠1 and ∠7; ∠2 and ∠8 are the alternate exterior angles.
- Same side Interior Angles: ∠3 and ∠5; ∠4 and ∠6 denote the interior angles on the same side of the transversal or co-interior or consecutive interior angles.

If the lines a and b are parallel to each other as shown, then the following axioms are given for angle pairs of these lines.

Axiom 1: Corresponding Angle Axiom

If two lines which are parallel are intersected by a transversal then the pair of corresponding angles are equal.

From Fig. 3: ∠1=∠6, ∠4=∠8, ∠2= ∠5 and ∠3= ∠7

The converse of this axiom is also true according to which if pair of corresponding angles are equal then the given lines are parallel to each other.

Theorem 1: If two lines which are parallel are intersected by a transversal then the pair of alternate interior angles is equal.

From Fig. 3: ∠4=∠5 and ∠3=∠6

Proof: As, ∠4=∠2 and ∠1=∠3(Vertically Opposite Angles)

Also, ∠2=∠5 and ∠1=∠6 (Corresponding Angles)

⇒∠4=∠5 and ∠3=∠6

The converse of the above theorem is also true which states that if the pair of alternate interior angles are equal then the given lines are parallel to each other.

Theorem 2: If two lines which are parallel are intersected by a transversal then the pair of interior angles on the same side of transversal are supplementary.

∠3+ ∠5=180° and ∠4+∠6=180°

As ∠4=∠5 and ∠3=∠6 (Alternate interior angles)

∠3+ ∠4=180° and ∠5+∠6=180° (Linear pair axiom)

⇒∠3+ ∠5=180° and ∠4+∠6=180°

The converse of the above theorem is also true which states that if the pair of co-interior angles are supplementary then the given lines are parallel to each other.

PARALLEL LINES IN REAL LIFE

One will be able to see lines which are parallel to each other in real life too, if only one has the patience and is observant enough to do so. For instance take the railroads. The railway tracks are literally parallel lines. The two lines or tracks are meant for the wheels of the train to travel along on. The difference between the parallel lines imagined by mathematicians and the ones who actually make the railway tracks is that, mathematicians have the liberty to imagine the parallel lines over flat surfaces and paper, while trains travel across all sorts of terrain, from hills, slopes and mountains to over bridges.