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Slope Criteria for Parallel & Perpendicular Lines

Instructor: David Karsner

David holds a Master of Arts in Education

If two lines have the same slope but different y-intercepts, they are parallel. If two lines have slopes that are negative reciprocals of each other, they are perpendicular. This lesson clarifies these definitions with a step-by-step explanation of how to recognize parallel and perpendicular lines in problems.

All Roads Lead to Rome

All Roads Lead to Rome
Rome Road Sign

If 'All roads lead to Rome' is true, all roads would have to intersect in Rome! But if those roads are parallel to each other, they will never intersect in Rome (or anywhere else). You can use the slope of lines to determine whether they intersect or are parallel. If they do intersect, you can use the slope to determine if they intersect perpendicularly. In this lesson, I will share some definitions related to lines, discuss how to determine if lines are parallel and perpendicular, and give a few examples.

Some Definitions

Parallel Lines are two or more lines that lie in the same plane that never intersect. They do not share any common points.

Perpendicular Lines are two lines that intersect so that their intersection forms four right angles.

Slope is a measure of how steep a line is. It is measured by finding how far one point of the line has to move vertically to reach another point, divided by how far that point has to move horizontally.

The Y-Intercept is the point of the graph that intersects with the y-axis.

The Negative Reciprocal of a number is that number, flipped upside down and given the opposite sign. For example, the number 1/3 has the negative reciprocal of -3/1, or just -3.

Skew Lines are two or more lines that are neither parallel nor intersecting. This only happens when the lines lie in different planes.

Parallel Lines

If two lines have the same slope, lie in the same plane, and are not the same line, the lines are parallel. To put it simply, if they have the same slope, they are parallel.

There are two exceptions to this simplification. One exception is for lines that are skew. Skew lines do not lie in the same plane, so they may have the same slope yet still not be parallel. The other exception is for two lines that are actually the same line. For, example the lines y=2x+1 and 4x-2y=-2 both have a slope of 2 -- but they are the same line, so we do not say that they are parallel.

To determine if two lines are parallel, you will need to discover the slope of both lines. There are several ways to find the slope of a line. You can determine the slope of a line by inspecting the graph, solving the equation for y, or by using two coordinate points from the graph.

Finding Slope from the Graph
Finding the Slope from the Graph

Finding the Slope from the Equation
Finding Slope from the Equation

Finding the Slope from Two Points
Finding Slope from Two Points

Perpendicular Lines

If the slopes of two lines are negative reciprocals of one another, then the two lines are perpendicular. Two numbers are reciprocals if their numerators and denominators have been switched. For example, 2/3 and 3/2 are reciprocals. To make that a negative reciprocal, the sign must change as well. For example, 2/3 and -3/2 are negative reciprocals. An integer, such as 4, can be written as a fraction, such as 4/1. 4 is the reciprocal of 1/4 and the negative reciprocal of -1/4.

To determine if two lines are perpendicular, you will need to determine the slope. You can find the slope using the graph, the points, or the equation, just as you can for parallel lines.

An Example or Two

Example 1.

Are the two lines given by y= (-2/3)x + 4 and 3x-2y=2 parallel, perpendicular or neither?

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