Are you interested in finding the parallel line equations and calculating the distance between two lines? If the answer is yes, check out our free online Parallel Line Calculator.

Besides it, would you like to explore more of our math calculators? If yes, we have various calculators, such as Area of a Trapezoid Calculator if you need to calculate the area of a trapezoid. Also, there is our Cross Product Calculator for calculating the product of vectors.

Parallel Lines – Definition

What are parallel lines in math, or more specifically in geometry?

In geometry, we define them as two lines located in the same plane (flat surface) at an equal distance from each other. Additionally, no matter how much you extend them, they will never meet. Therefore, parallel lines are central and always form the same slope.

If two lines are parallel, then we use the symbol || to represent their relationship. So, for example, AC || DE denotes that the line AC is alongside to the line DE. 

We find them useful mainly in understanding the relationships between the paths of objects and sides of various shapes.

Properties

According to the parallel lines definition in math, we conclude that each of them share the same properties. In other words, they always have these properties:

  • Vertically opposite angles of two parallel lines are equal
  • Corresponding angles in parallel lines are equal
  • Alternate exterior angles are equal
  • Alternate interior angles are equal

Slope

A parallel line never intersect with another; therefore, they are pointing in an infinite direction. This direction where they are going is called the slope of parallel lines

For example, you are given two points, A(x1, y1) and B(x2, y2), on a line m. How to find the slope of the line m? We can calculate it with the simple equation:

Slope=\frac{y_{2} - y_{1}}{x_{2} - x_{1}} 

What is the slope of a parallel line?

Calculating the slope for a parallel line is almost the same as for a simple line. Since we have two alongside lines, both of them have two points that they pass through:

1st line – A(x1, y1) B(x2, y2) and C(x1, y1) D(x2, y2)

2nd line – F(x1, y1) G(x2, y2) and H(x1, y1) I(x2, y2)

So, to calculate the slope for them, we use the same equation:

Slope=\frac{y_{2} - y_{1}}{x_{2} - x_{1}} 

You will get the same result for both 1st line and 2nd line.

Parallel Line Equation

Every straight line in a 2D coordinate system can be written as an equation below:

y = ax + b

where:
y – y-axis intercept
a – a coefficient
x – x-axis intercept
b – b coefficient

Parallel Lines and Transversals

In what relationship are parallel lines with a transversal?

For instance, if we consider two parallel lines and draw another line that intersects both, that line is called a transversal. Additionally, if two parallel lines are cut by a transversal, they form eight different angles.

In comparison, perpendicular lines always intersect at a 90-degree angle.

Distance Between Two Parallel Lines: How to Find?

We can find or measure the distance (shortest segment that would join them) of the parallel lines formula below:

D = \frac{\left| b - r \right|}{\sqrt{m^{2} + 1}}

Note: You can easily find the distance between two lines using our calculator if you don’t want to use formulas or calculations. You can leave all of that heavy work to our calculator and instantly get the result back to you.

Parallel Line Calculator – How to Use?

One of the things that our calculator calculates is the distance between two lines. However, if you are interested in finding it manually, head to the previous section to see the formula.

But, besides that, our calculator helps you find the equation of a line parallel to the first line you specify without problems. Therefore, you will learn how to calculate both using our calculator. Thus, follow the steps below.

How to find the equation of a parallel line and calculate the distance that they form:

  • Enter the variables for the equation of the first line (y = mx + r)
  • Enter the coordinates of a point through which the second line passes – For example, A (1, 4)
  • The calculator will return you the new line equation and the distance between them

Parallel Line Calculator – Example

In the previous part, you learned everything you need to know to use our calculator to find the line equations and calculate the distance between them. Therefore, this section will show those steps in a real example.

Scenario: You are given an equation of a line y = 3x – 2 and a point A (-2, 3) through which another line is passing. How to find the equation of a parallel line and the distance between them?

Steps:

  1. Enter the variables (m and r) in the calculator. In our case, m = 3 and r = -2
  2. Enter the coordinates of a point A in the calculator
  3. The calculator returns the new line equation: y = 3x + 9
  4. Additionally, you will also get the distance between the two lines: 3.4785

Parallel Lines in Real Life

Are you interested in some of the parallel lines examples in real life? Finding parallel lines used in real-world situations is n because this math concept is undoubtedly important.

A pedestrian crossing - zebra's white stripes - Real world parallel lines.
A pedestrian crossing – zebra’s white stripes alongside to each other

Let’s check out some of the real-world usages of them:

  • Railway tracks – the iron bars lay alongside to each other
  • A pedestrian crossing – zebra’s white stripes are alongside to each other
  • Piano keys
  • Table – each pair of the opposite sides of a table are alongside to each other

Note: If you need to calculate the distance between the two lines, feel free to utilize our Parallel Line Calculator. You can check it out on our website or through a mobile app.

FAQ

How do you determine if two lines are parallel?

In order to determine whether two lines are parallel, we need to compare their slope. So, if their slopes are found to be equal, it indicates that the two lines are parallel.

Do parallel lines have the same slope?

Yes, each pair of parallel lines have the same slope.

Are parallel lines coplanar?

Yes, they are always coplanar, and they never intersect.

How to find the slope of a parallel line?

Calculating the slope for a parallel line is almost the same as for a simple line. So, for example, you are given two points, point A(x1, y1) and point B(x2, y2), on a line m.
We can calculate the slope utilizing the simple equation:
(y2y1) / (x2x1)

How many solutions do parallel lines have?

Any parallel line, by default, has no solutions.

What is the parallel line equation of a line y = 3x + 2 and a line that passes through point A(7,8)?

You can calculate it using the standard formula for finding the parallel line equation and the distance between the lines. However, we recommend you use our simple and quick calculator that will solve it; find the new equation and the distance.
Regardless of the method you choose, the new equation of a line you get eventually is y = 3x – 13.
The distance equals 4.743.

How many parallel lines does a trapezoid have?

According to the definition, trapezoids are geometric shapes with only one pair of opposite sides parallel. However, some people consider that trapezoids have at least one pair of opposite sides parallel, which means they can have more than one pair.