Parallel and Perpendicular Lines – Definition With Examples
Updated on February 11, 2026
Understanding how lines relate to each other makes sense because lines are all around us. When we stay at home, or go outside, in the books, buildings, and stores. Today, we meet two of the most important relationships: parallel lines and perpendicular lines.
In this article, Brighterly explores with ease all the questions around parallel and perpendicular lines: what is parallel lines, parallel and perpendicular lines equations, etc.
What are parallel and perpendicular lines?
Parallel lines are lines that never touch and always stay the same distance apart, like the rails on a train track. This is how parallel definition math says.
Perpendicular lines are lines that cross each other at a right angle (90°), like the corner of a book or a table. In math with coordinates, parallel lines have the same slope, and perpendicular lines have slopes that multiply to –1. One line can be parallel to one line and perpendicular to another. When lines are perpendicular, they make four right angles where they meet. If a line crosses parallel lines, it makes matching or alternate angles.
Definition of parallel lines
Parallel lines share the same gradient when they are two (or more) straight lines.
The distance between any two straight lines remains constant in both directions, and each line has the same gradient. As a result, all lines are parallel.
Therefore, the equation of a line parallel to a line with gradient 2 can be written in the form y = 2x + c, where c is the y-intercept.
Definition of perpendicular lines
A line perpendicular to another has a gradient that multiplies with the other line’s gradient to give −1. This means that the gradients of two perpendicular lines are negative reciprocals of each other. That’s what the definition of perpendicular lines in geometry says.
As an example, let’s look at the line y = x to see why:
When we move right along the x-axis, we move one unit square up the y-axis, so m = 1 is the gradient of y = x.
Next, let’s construct a line perpendicular to y = x. For example, the line y = −x is perpendicular to y = x.
An equation for a straight line perpendicular to y = x is y = −x.
There is a difference in the gradients of the two lines (one has a positive gradient, the other a negative one). Also, we need to learn some things about the slope of parallel and perpendicular lines.
Properties of parallel and perpendicular lines
The following are some of the key properties of perpendicular and parallel lines:
Parallel and perpendicular lines follow specific rules that help us identify and work with them in geometry. These properties are true in both shapes and coordinate planes.
Properties of parallel lines
Parallel lines possess the following properties:
- Parallel lines never meet, no matter how far they are extended.
- Parallel lines always stay the same distance apart.
- Lines that are parallel have the same slope in the coordinate plane.
- Parallel lines do not form any angles where they run side by side.
- A line that cuts across parallel lines makes matching (corresponding) and opposite (alternate) angles.
- Parallel lines can extend infinitely in both directions.
Properties of perpendicular lines
Perpendicular lines have the following properties:
- Perpendicular lines cross each other at a right angle (90°).
- Perpendicular lines make four right angles.
- In the coordinate plane, the slopes of perpendicular lines multiply to –1.
- One line can be perpendicular to multiple lines at different points.
- Perpendicular lines can create squares, rectangles, and other right-angled shapes.
- Perpendicular lines are used to measure and divide space accurately in geometry and real life.
How to find parallel and perpendicular lines?
You can tell if lines are parallel or perpendicular by looking at their slopes or angles. Parallel lines have the same slope. This means they go in the same direction and never meet. If the slopes are different, the lines are not parallel.
Perpendicular lines cross at a right angle (90°). In the coordinate plane, the slopes of perpendicular lines multiply to –1. For instance, when one line has a slope of 2, a perpendicular line will have a slope of −1/2.
You can also check the angles where the lines meet. If the lines make a right angle, they are sure perpendicular. If the lines never meet and stay the same distance apart, they are parallel.
Difference between parallel and perpendicular lines
Perpendicular and parallel lines are different! They can be distinguished by comparing their gradients (slopes).
| Relationship Between Lines | Gradient Rule | Example Equations | |
Parallel Lines |
Lines never meet, always the same distance apart | Gradients are equal (m₁ = m₂) | y = 2x + 3 and y = 2x − 5 |
Perpendicular Lines |
Lines intersect at a right angle (90°) | Gradients multiply to −1 (m₁ × m₂ = −1) | y = 3x + 1 and y = −1/3x + 2 |
Perpendicular lines in real life
Perpendicular lines are lines that intersect at a 90° angle. They are common in both nature and human-made structures. Here are some simple examples of perpendicular lines in everyday life:
- The corner of a book or notebook where the edges meet.
- The intersection of two streets in a city grid.
- The edges of a window frame or door frame.
- The legs of a table or chair meeting the tabletop.
- A streetlight pole standing upright on the sidewalk.
- The cross on top of a church steeple.
- The beams in a ceiling forming a rectangular pattern.
Parallel lines examples
Perpendicular lines examples in real life can be described as follows:
- The floor and walls of a room form perpendicular lines.
- The hands of a clock at 3:00 or 9:00 form a right angle.
- A flagpole standing upright on flat ground.
- The corners of a rectangular picture frame are perpendicular.
Now, let’s learn for writing equations of parallel and perpendicular lines.
Equations of parallel and perpendicular lines
The equation of a parallel line uses the same slope as the original line. For example, if a line is y=2x+3, a parallel line could be y=2x−5.
The equation of a line perpendicular to another line has a slope equal to the negative reciprocal of that line’s slope. For example, if a line is y=2x+1, a perpendicular line will have the slope −1/2, like y=−1/2x+4.
To write these equations, you can also use a point on the line with the point-slope formula: y−y1=m(x−x1), where m is the slope and (x1,y1) is a point on the line.
As you can see, we can write equations of parallel or perpendicular lines using the slope of the given line and the point-slope formula.
Writing equations of parallel lines
Identifying parallel lines can be done by checking the gradients of their equations. Some rearranging may be necessary.
As an example, show that y = 3x − 5 and 6x − 2y = 8 are parallel.
Gradient 3 is the gradient of the first line. To find the gradient, we must rearrange the second equation into the form y = mx + c.
6x − 2y = 8
−2y = −6x + 8
y = 3x − 4
There is also a gradient of 3 on the second line. Both lines have the same gradient, so they are parallel.
Writing equations of perpendicular lines
Perpendicular lines can be identified by checking the product of their gradients. If two lines are perpendicular, the product of their gradients is -1 (one gradient is the negative reciprocal of the other).
As an example, show that y = 2x + 3 and x + 2y = 10 are perpendicular.
Gradient 2 is the gradient of the first line (m_1 = 2). To find the gradient of the second line, we must rearrange the equation into the form y = mx + c:
x + 2y = 10
2y = -x + 10
y = – 1/2 x + 5
The gradient of the second line is -1/2(m2 = -1/2).
To check if they are perpendicular, multiply the gradients:
2x (-1/2) = -1
The product of the gradients is -1, so the lines are perpendicular.
Practice problems on parallel and perpendicular lines
- Two streets meet at a corner forming a 90° angle. Are these streets perpendicular?
- The opposite edges of a rectangular table are measured. Will these edges ever meet if extended?
- A notebook has lines drawn for writing. Are the lines parallel or perpendicular?
- Two lines have slopes m1= 3 and m2 = 4. Are these lines parallel?
- A line has the equation y = 2x + 5. Write the equation of a line perpendicular to it.
Conclusion
We learned how lines can be connected in two main ways. We also know that in coordinate geometry, the slopes of parallel and perpendicular lines are different. Parallel lines have the same slope. Perpendicular lines have slopes that multiply to -1. One line can be parallel to one line and perpendicular to another. Perpendicular lines form four right angles. Parallel lines can be crossed by another line, making matching or alternate angles. By learning this, young students can master drawing lines, solving geometry problems, and understanding shapes.
Frequently asked questions on parallel and perpendicular lines
What is the difference between parallel and perpendicular lines?
Parallel lines never touch each other and stay on the same distance, while perpendicular lines cross at a right angle (90°). Parallel lines have the same slope, while perpendicular lines multiply to –1.
How to find parallel and perpendicular lines?
To find parallel lines, check the slope of parallel lines and perpendicular lines. To find perpendicular lines, check if the slopes multiply to –1. You can also look at the angles: if two lines meet at 90°, they are perpendicular. Lines that never meet are parallel.
Can a pair of lines be both parallel and perpendicular?
No. Lines cannot be both parallel and perpendicular. Parallel lines never meet, but perpendicular lines must cross at a right angle. A line cannot stay apart and cross at the same time.
How do you tell if a line is parallel or perpendicular?
Check the slopes or angles. Lines with the same slope are parallel. Lines with slopes that multiply to –1 are perpendicular. You can also see if the lines meet: a 90° angle means perpendicular, never meeting means parallel.
Parallel and perpendicular lines worksheet
To master your lines knowledges use Brighterly’s worksheets
- Parallel and Perpendicular Lines Worksheets
- Lines and Angles Worksheets
- Parallel Lines and Transversals Worksheets
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