# Parallel Lines – Definition, Properties, and Examples

When working with parallel lines, it is important to be familiar with their definition and properties. Let’s go ahead and begin with its definition.

Parallel lines are lines that are lying on the same plane but will never meet.

Understanding what parallel lines are can help us find missing angles, solve for unknown values, and even learn what they represent in coordinate geometry.

Since parallel lines are used in different branches of math, we need to master them as early as now.

## What are the parallel lines?

Parallel lines are equidistant lines (lines having equal distance from each other) that will never meet.
These are some examples of parallel lines in different directions: horizontally, diagonally, and vertically.

Another important fact about parallel lines: they share the same direction.

### What are some real-world examples of parallel lines?

• Roadways and tracks: the opposite tracks and roads will share the same direction, but they will never meet at one point.
• Lines on a writing pad: all lines are found on the same plane, but they will never meet.
• Pedestrian crossings: all painted lines lie along the same direction and road, but these lines will never meet.

### How do we use parallel lines in coordinate geometry?

• When the graphs of two linear equations are parallel in coordinate geometry, the two equations do not share a solution.
• The slopes of two parallel lines are equal in coordinate geometry.

## How to prove lines are parallel?

Several geometric relationships can be used to prove that two lines are parallel.

Before we begin, let’s review the definition of transversal lines.

Transversal lines are lines that cross two or more lines

The image shown to the right shows how a transversal line cuts a pair of parallel lines

When a transversal line cuts a pair of parallel lines, different pairs of angles are formed. These different types of angles are used to prove whether two lines are parallel to each other.

In the next section, you’ll learn what the following angles are and their properties:

•  Corresponding angles
• Alternate interior angles
• Alternate exterior angles
• Consecutive interior angles

## Properties of parallel line

When a transversal line cuts two lines, the properties below will help us determine whether the lines are parallel.

1. Two lines cut by a transversal line are parallel when the corresponding angles are equal.

The two pairs of angles shown above are examples of corresponding angles. In general, they are angles that are in relative positions and lying along the same side.

2. Two lines cut by a transversal line are parallel when the alternate interior angles are equal.

Alternate interior angles are a pair of angles found in the inner side but are lying opposite each other.

3. Two lines cut by a transversal line are parallel when the alternate exterior angles are equal.

Alternate exterior angles are a pair of angles found on the outer side but are lying opposite each other.

4. Two lines cut by a transversal line are parallel when the sum of the consecutive interior angles is $\boldsymbol{180^{\circ}}$.

Consecutive interior angles are consecutive angles sharing the same inner side along the line.

5. Two lines cut by a transversal line are parallel when the sum of the consecutive exterior angles is $\boldsymbol{180^{\circ}}$.

Consecutive exterior angles are consecutive angles sharing the same outer side along the line.

### Summary of parallel lines definition

Let’s summarize what we’ve learned so far about parallel lines:

• they are coplanar lines
• they are equidistant
• they will never meet

The properties below will help us determine and show that two lines are parallel.

1.  Corresponding angles are equal.

Example: $\angle b ^{\circ} = \angle f^{\circ}, \angle a ^{\circ} = \angle e^{\circ}e$

2. Alternate interior angles are equal.

Example: $\angle c ^{\circ} = \angle f^{\circ}, \angle d ^{\circ} = \angle e^{\circ}$

3. Alternate exterior angles are equal.

Example: $\angle a ^{\circ} = \angle h^{\circ}, \angle b^{\circ} = \angle g^{\circ}$

4. Consecutive interior angles add up to $180^{\circ}$.

Example: $\angle c ^{\circ} + \angle e^{\circ}=180^{\circ}$, $\angle d ^{\circ} + \angle f^{\circ}=180^{\circ}$

5. Consecutive exterior angles add up to $180^{\circ}$.