Parallel Lines

In this tutorial, we will explore the concept of parallel lines and the relationships between the angles formed when a transversal cuts them. A good understanding of these is essential if you want to be good at geometry.

So let’s dive in.

Parallel Lines Cut by a Transversal

Two lines are said to be parallel if they are in the same plane and are always at the same distance from one another (hence, they never intersect).

Now, a line that intersects (or cuts) two lines in the same plane at two distinct points is known as a transversal.

Pairs of Angles

When a transversal cuts two parallel lines, a total of eight separate angles are formed – a through f in the figure below.

From these eight angles, there are certain pairs of angles with special relationships connecting them – they are either equal or their sum is $\hspace{0.2em} 180 \degree \hspace{0.2em}$.

Let’s explore the angle pairs in greater detail.

Alternate Interior Angles

A pair of angles that lie between the parallel lines and on opposite sides of the transversal form a pair of alternate interior angles. The two angles in each pair are equal.

There are two pairs of alternate interior angles.

Alternate Exterior Angles

A pair of angles that lie outside the parallel lines and on opposite sides of the transversal form a pair of alternate exterior angles. The two angles in each pair are equal.

There are two pairs of alternate exterior angles.

Corresponding Angles

A pair of angles that lie on the same side of the transversal as well as the same sides of the parallel lines (above or below) make a pair of corresponding angles. The two angles in each pair are equal.

There are four pairs of corresponding angles.

Consecutive Interior Angles

Two angles lying between the parallel lines and on the same side of the transversal make a pair of consecutive interior angles. The sum of two angles in each such pair is 180o.

There are two pairs of consecutive interior angles.

Vertically Opposite Angles and Linear Pairs

Vertically Opposite Angles

When two straight lines intersect as shown in the figure below, there are four angles formed around the point of intersection.

The two pairs of opposite angles are known as vertically opposite angles. Angles in each pair are equal.

Linear Pairs

When two lines intersect, each pair of adjacent angles are supplementary – have a sum of $\hspace{0.2em} 180 \degree \hspace{0.2em}$ – and are known as linear pairs of angles.

Example

$\hspace{0.2em} L_1 \hspace{0.2em}$ and $\hspace{0.2em} L_2 \hspace{0.2em}$ are parallel lines. If $\hspace{0.2em} q = 65.4 \degree \hspace{0.2em}$, find the values of $\hspace{0.2em} p \hspace{0.2em}$, $\hspace{0.2em} r \hspace{0.2em}$, $\hspace{0.2em} s \hspace{0.2em}$, and $\hspace{0.2em} t \hspace{0.2em}$ and mention the relationship of these angles with $\hspace{0.2em} q \hspace{0.2em}$.

Solution

Let’s start with $\hspace{0.2em} p \hspace{0.2em}$ and $\hspace{0.2em} q \hspace{0.2em}$. They are a linear pair, meaning they must add up to $\hspace{0.2em} 180 \degree \hspace{0.2em}$. So,

Substituting the value of $\hspace{0.2em} q \hspace{0.2em}$, we have –

Now, since $\hspace{0.2em} L_1 \hspace{0.2em}$ and $\hspace{0.2em} L_2 \hspace{0.2em}$ are parallel, we can use the properties of the pairs discussed above to our advantage.

$\hspace{0.2em} q \hspace{0.2em}$ and $\hspace{0.2em} r \hspace{0.2em}$ are consecutive interior angles. So they would be supplementary (sum $\hspace{0.2em} = 180 \degree \hspace{0.2em}$).

$\hspace{0.2em} q \hspace{0.2em}$ and $\hspace{0.2em} s \hspace{0.2em}$ are alternate interior angles and hence, they must be equal.

Finally, $\hspace{0.2em} q \hspace{0.2em}$ and t are corresponding angles. So they would be equal too.

How to Check if Two Lines Are Parallel

If we have two lines cut by a transversal, we might be able to determine whether the lines are parallel – depending on our knowledge of the angles formed.

Here’s how.

For example, if the lines are parallel, corresponding angles must be equal.

Example

For each of the following figures, find if lines $\hspace{0.2em} L_2 \hspace{0.2em}$ and $\hspace{0.2em} L_2 \hspace{0.2em}$ are parallel.

Solution (a)

If we assume the lines are parallel, the two marked angles ($\hspace{0.2em} 64.2 \degree \hspace{0.2em}$ and $\hspace{0.2em} 64.5 \degree \hspace{0.2em}$) would be alternate interior angles. And hence, they would also be equal. However, they are not.

So we end up with a contradiction. And so, the lines are not parallel.

Solution (b)

This is a slightly tricky one. But before we proceed, let me redraw the figure and name some angles so it becomes easier to explain.

The pair of angles marked in the original figure ($\hspace{0.2em} a \hspace{0.2em}$ and $\hspace{0.2em} b \hspace{0.2em}$) do not fall into any of the three categories discussed above. So based on them alone, we cannot determine whether the lines are parallel.

But if we can find angle $\hspace{0.2em} c \hspace{0.2em}$, we will have a pair of corresponding angles – $\hspace{0.2em} a \hspace{0.2em}$ and $\hspace{0.2em} c \hspace{0.2em}$. And that we can use for our test.

We can see that $\hspace{0.2em} b \hspace{0.2em}$ and $\hspace{0.2em} c \hspace{0.2em}$ are a linear pair (regardless of whether the lines are parallel).

Now that we have $\hspace{0.2em} c \hspace{0.2em}$, we can see that $\hspace{0.2em} a \hspace{0.2em}$ and $\hspace{0.2em} c \hspace{0.2em}$ (a pair of corresponding angles) are equal. This is consistent with our assumption that the lines are parallel.

So yes, the lines are parallel.

And with that, we come to the end of this tutorial on parallel lines cut by a transversal. Until next time.