Parallel Lines

The two straight lines which maintain equidistance at all their opposite points and never intersect each other in a plane are defined parallel lines.

Two straight lines are often appeared side by side in a plane. Sometimes, they have some properties commonly. Parallelism is case of two straight lines in which straight lines have four properties commonly.

- Equal distance at all their opposite points.
- Never intersect each other at any point.
- Separate on a plane.
- Have same angle.

If two straight lines have the four properties, one straight line is known as a parallel line to another and vice-versa. Therefore, the two straight lines are called parallel lines.

Parallelism of two straight lines is denoted by the symbol ($\parallel $) in geometric system.

Equidistance at all opposite points

$\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are two straight lines. Let us check the parallelism of these two straight lines.

- The straight lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ both are in same plane.
- The angle made by the $\overleftrightarrow{AB}$ is zero degrees and the angle made by the $\overleftrightarrow{CD}$ is also zero degrees. In other words, both straight lines maintains same angle in the plane.
- The distance from the line $\overleftrightarrow{AB}$ to line $\overleftrightarrow{CD}$ is $d$ units. The distance from every point on line $\overleftrightarrow{AB}$ to exact opposite point on the line $\overleftrightarrow{CD}$ is same.
- Due to equidistance and having same angles, the lines $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ do not intersect each other at any point in the plane.

Therefore, $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ are parallel lines. The line $\overleftrightarrow{AB}$ is parallel to the line $\overleftrightarrow{CD}$ and it is written as $\overleftrightarrow{AB}\parallel \overleftrightarrow{CD}$ in mathematics. Similarly, the line $\overleftrightarrow{CD}$ is also parallel to the line $\overleftrightarrow{AB}$ and it is written as $\overleftrightarrow{CD}\parallel \overleftrightarrow{AB}$ in mathematics.

Let us look at another example. There are two lines on a plane and they are called lines $\overleftrightarrow{PQ}$ and $\overleftrightarrow{RS}$. Now, check the parallelism of straight lines geometrically.

Equal Angles by the Parallel Lines

- The straight lines $\overleftrightarrow{PQ}$ and $\overleftrightarrow{RS}$ are in same plane.
- The straight line $\overleftrightarrow{PQ}$ makes an angle ${30}^{\xb0}$ and $\overleftrightarrow{RS}$ is also making same angle in the plane.
- The distance between lines $\overleftrightarrow{PQ}$ and $\overleftrightarrow{RS}$ is $m$ units and the distance between them is unique at their all opposite points.
- Both lines are making same angle and the distance between them is unique at all the opposite points. Therefore, the lines $\overleftrightarrow{PQ}$ and $\overleftrightarrow{RS}$ never intersect each other at any point on the plane.

The straight lines $\overleftrightarrow{PQ}$ and $\overleftrightarrow{RS}$ are called parallel lines geometrically. Therefore, it is expressed as $\overleftrightarrow{PQ}\parallel \overleftrightarrow{RS}$ or $\overleftrightarrow{RS}\parallel \overleftrightarrow{PQ}$.

There is one important phenomena involved in the concept of parallelism of straight lines. If the two lines are in same plane and make same angle, the distance between them is same at all the opposite points of both lines and they never either touch or intersect each other at any point in the plane.

In other words, if distance between all opposite points of two lines is same, they make same angles because angles of parallel lines and distance between them have a direct relation geometrically.

So, just check either they are making same angles or they are maintaining equidistance at all the opposite points of the both lines to verify the parallelism of two straight lines in a plane in the geometric system.