Parallel and perpendicular lines are two key concepts in geometry. Here are the definitions of parallel and perpendicular, a look at their properties, and how to use slope to identify them.
Parallel lines are lines that never cross each other (intersect) and always stay the same distance apart. They share 0 points in common with each other. Two different parallel lines have the same slope as each other.
Properties of Parallel Lines
- In the same plane
- Never intersect
- Remain the same distance apart
- Have the same slope as each other
- Symbol is ||
Examples of Parallel Lines
Here are examples of parallel lines and line segments:
- The paths of cars travelling in two lanes
- The parallel sides of a square, rhombus, rectangle, or parallelogram
- Railroad tracks
- The rungs of a ladder
- The lines on ruled paper
Perpendicular lines cross each other at exactly one point, making a 90° angle (right angle) with each other. Like parallel lines, perpendicular lines exist in the same plane as each other (coplanar). The product of the slopes of two perpendicular lines is -1.
Properties of Perpendicular Lines
- In the same plane
- Intersect at one point
- Intersect at a 90°
- Slope of one line is m and slope of the other line is -1/m (product of their slopes is -1)
- Symbol is ⊥
Examples of Perpendicular Lines
Here are examples of perpendicular lines, line-segments, and planes in daily life:
- The intersecting sides of squares or rectangles
- The line segments in the letters “T” and “L”
- The legs of a right triangle
- The stripes on the flag of Norway
- The walls and floors of a room
Can a Pair of Lines Be Both Parallel and Perpendicular?
No, a pair of lines cannot be both parallel and perpendicular. The lines can be parallel, perpendicular, or else intersecting yet not perpendicular.
Practice Identifying Parallel and Perpendicular Lines
Download or print this free math worksheet for practice identifying parallel, perpendicular, and intersecting lines that are not perpendicular. Just select the appropriate download link for your needs.
Using Slope to Identify Parallel and Perpendicular Lines
Compare the equations of two lines and identify whether they are parallel or perpendicular. The slope-intercept equation of a line is y = -mx + b, where x and y identify a point, m is the slope, and b is y-intercept.
- Two parallel lines have the same slope, but different y-intercepts. m1=m2, where m1 and m2 are the slopes of two parallel lines.
- Two perpendicular lines have slopes m and -1/m. A quick check to see if the lines are perpendicular is if the product of their slopes equal -1 (m1 x m2 = -1).
So, the slope or “m” is the same for parallel lines. For example, two lines with equations y = -3x +6 and y = -3x -4 have the same slope (3), so you know they are parallel lines. Be careful that two lines are not, in fact, the same line! If both the slope and the y-intercept are the same, the you are dealing with one line written two different ways. For example, y = 3x + 2 and y -2 = 3x represent two ways of writing the exact same equation.
Perpendicular lines have different slopes from each other. The slope of one line is the negative reciprocal of the other (m1 = m and m2 = -1/m) . The product of their slopes is -1 (m1 x m2 = -1). For example, the lines y = 1/4x + 3 and y = -4x + 2 are perpendicular because you can see one slope is the negative reciprocal of the other.
So, are these two lines parallel or perpendicular?
y = 2x + 1
y = -0.5x + 4
First, identify the slopes of the lines. For the first equation, the slope is 2. The slope of the second equation is -0.5. These two values are not the same, so you know the lines are not parallel.
Next, see whether or not the lines are perpendicular. Check this by multiplying the slopes of the lines.
2 x (-0.5) = -1
The product of the slopes is -1, so the two lines are perpendicular.
Lines That Are Neither Parallel nor Perpendicular
Lines that intersect at any angle besides 90° are neither parallel nor perpendicular. These lines have different slopes from each other. An example of lines that are neither parallel nor perpendicular are the hands of a clock at 12 and 4.
- Altshiller-Court, Nathan (1925). College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (2nd ed.). New York: Dover Publications, Inc.
- Kay, David C. (1969). College Geometry. New York: Holt, Rinehart and Winston.
- Richards, Joan L. (1988). Mathematical Visions: The Pursuit of Geometry in Victorian England. Boston: Academic Press. ISBN 0-12-587445-6.