Parallel Lines

Parallel lines definition

Parallel lines are the types of lines that consists of two or more lines in the same plane.

Parallel lines remain at the same distance from each other, no matter how far they are extended. They can be constructed in any direction whether horizontal, vertical, or diagonal. Mathematically, they are represented with the symbol $\parallel$ which is called “ is parallel to”.

Parallel lines, StudySmarter Originals

Here in the above figure p and q are parallel lines and m and n are parallel lines. Hence it is said that$p\parallel q$and$m\parallel n.$But line a and line b are not parallel to each other as when extending both the lines, they both will intersect each other at some point. So a is not parallel to b (that is$a\nparallel b$).

Parallel lines angles

As parallel lines do not intersect with each other, no angles can be formed between them. But when another line apart from the given parallel lines intersect both the parallel lines then some angles are formed between them.

Transversal cutting parallel lines, StudySmarter Originals

Here in the above figure, we can see that line l cuts both the parallel lines a and b. So line l is the transversal line. As the transversal cuts parallel lines it can be seen that the transversal forms pairs of angles with both the lines. There are different types of angles created by transversals.

Corresponding angles

Corresponding angles can be easily identified in the form of an “F” shape. They can be formed anyway either upside down or back and front. Corresponding angles are always equal to each other in parallel lines.

Corresponding angles of parallel lines, StudySmarter Originals

Alternate angles

Alternate angles can be found in the form of a “Z” shape. They can be both interior and exterior angles. Similarly, like corresponding angles, alternate angles can be formed in any direction. Pair of alternate angles are always equal to each other.

Alternate interior angles of parallel lines, StudySmarter Originals

Here in the above figure, both the angles are alternate interior angles.

Interior angles

Interior angles are formed in the shape of a “U”. They can be found on either side of the transversal containing both parallel lines. The sum of interior angles will always be $180°.$

Interior angles of parallel lines, StudySmarter Originals

Exterior angles

Exterior angles are formed in the shape of a “U” but will be located on the outside region of it. And the sum of the pairs of exterior angles will always be$180°.$

Exterior angles, StudySmarter Originals

Vertically Opposite angles

Vertically opposite angles are found in the form of two “V” touching each other. They only contained any one of the parallel lines for each pair. Vertically opposite angles are equal to each other.

Vertically Opposite angles, StudySmarter Originals

So we can represent all the pairs of angles for all the types of angles as below.

All pairs of angles in parallel lines, StudySmarter Originals

• Corresponding angle pairs :$\angle A\&\angle E;\angle B\&\angle F;\angle C\&\angle G;\angle D\&\angle H$
• Alternate angle pairs : $\angle C\&\angle E;\angle D\&\angle F$
• Interior angle pairs :$\angle C\&\angle F;\angle D\&\angle E$
• Exterior angle pairs : $\angle A\&\angle H;\angle B\&\angle G$
• Vertically opposite angle pairs : $\angle A\&\angle C;\angle B\&\angle D;\angle E\&\angle G;\angle F\&\angle H$

Parallel line equations

Parallel lines are one type of line. So we can represent parallel lines in the form of an equation of the line. We know that in coordinate geometry, the equation of the line can be written in the form of $y=mx+b.$ So we can also represent parallel lines in the form of the equation $\mathit{y}\mathbf{=}\mathit{m}\mathit{x}\mathbf{+}\mathit{b}.$

Here b is the y-intercept, so it can be any value. It is important to remember that as we have two or more lines in parallel lines the value of b for every line should be different from each other. As if they are equal then the equations of all lines will be the same and then it can be considered as one single line.

And m is the gradient or slope of that line. Here contrary to b the value of m for all the parallel lines should be equal. As m represents the slope of the line, if m is different for all parallel lines, then they will intersect each other and would not be considered parallel anymore.

We will understand the concept of gradient and how it can be found soon in the following topic.

Parallel lines equation in graph, StudySmarter Originals

Parallel lines gradient

The gradient or slope of parallel lines is the steepness of that line in the graph. The gradient of parallel lines is calculated with respect to the positive x-axis of the graph and parallel lines are inclined with the positive x-axis.

We know from above that the equation for parallel lines is $y=mx+b.$ Now suppose that the equation for one line $y=m_{1}x+b_1$and the equation for the other line is $y=m_{2}x+b_2.$ Here$b_1,b_2$ are y-intercept and m is the gradient of parallel lines. Then for both the lines to become parallel, the slope of both the lines should be equal. That is id="2715519" role="math" ${\mathit{m}}_{\mathbf{1}}\mathbf{=}{\mathit{m}}_{\mathbf{2}}.$ This equality can be derived by considering the angle between both the lines.

If we are already given two points on each line of the graph then we can calculate and verify the slope using the formula:

$\mathit{m}\mathbf{=}\frac{{\mathbf{y}}_{\mathbf{2}}\mathbf{-}{\mathbf{y}}_{\mathbf{1}}}{{\mathbf{x}}_{\mathbf{2}}\mathbf{-}{\mathbf{x}}_{\mathbf{1}}},$ where$x_1,x_2,y_1,y_2$are the x-axis and y-axis points for the single line.

Parallel lines examples

Let us see some parallel line examples and understand how to find angles and the slope in parallel lines.