Perpendicular Angles: Definition & Examples

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Perpendicular lines and surfaces show up all the time in the world around us. This lesson will allow us to become familiar with these concepts. After you have finished the lesson, you can test your knowledge with a quiz.

Right Angles

Angles can have any measure up to 360 degrees. In this lesson, we are concerned with a particular type of angle, and that is the right angle. Right angles are angles that measure 90 degrees. These types of angles show up very often in the world around us. For instance, if you are in a room with a flat ceiling, the angle between a wall and the ceiling is 90 degrees, so it is a right angle. To indicate a right angle, we use a box as shown in the following image.

Right Angles
perpendicular angle 8

Perpendicular

Now that we know what right angles are, we can concentrate on how they are formed. In general, angles are formed when two lines or surfaces intersect. For instance, in our ceiling and wall example the two surfaces intersect to create an angle.

When two lines or surfaces intersect to make a right angle, we say they are perpendicular. To see this, observe the following images.


Perpendicular Surfaces
perpendicular angles 6


Perpendicular Lines
perpendicular angles 10


Perpendicular Lines
perpendicular angles 9


The image of the building shows perpendicular surfaces, because when one edge of the window meets the other edge of the window, they form a right angle. The traffic intersection sign shows two perpendicular lines. Where the two line segments meet, a right angle is formed. Lastly, The lines shown on the running track form right angles where they intersect, so the lines are perpendicular.

These are just a few examples of perpendicular lines and surfaces in the world around us. Most likely, you can look up from your computer and see quite a few right angles in the room around you. The intersecting lines or surfaces that form those angles are perpendicular.

Slopes of Perpendicular Lines

As a quick reminder, the slope of a line is how quickly the line is rising or falling. When the slope of a line is positive, the line is rising from left to right, and when the slope is negative, the line is falling from left to right. The slopes of perpendicular lines have a special relationship. Slopes of perpendicular lines are negative reciprocals of each other.

The negative reciprocal of a number is found by flipping the number's numerator and denominator, and then switching the sign from positive to negative or negative to positive. An easy way to remember this is 'flip and switch'. For example, the negative reciprocal of the number 1/2 is found by flipping the numerator and denominator to get 2/1, or 2, and then switching the sign from positive to negative to get -2. Thus, the negative reciprocal of 1/2 is -2.

Okay, back to our perpendicular lines. As we said, the slopes of perpendicular lines are negative reciprocals of each other. The flipping of the numerator and denominator and the switching of the sign of the slope of one line resituates the line so that it falls perpendicular to the original line. Thus, if a line has slope a / b, then to find the slope of any line that is perpendicular to that line, we flip and switch a / b to get -b / a.


Perpendicular Lines
perpendicular angles 3


For instance, suppose a line has slope 4. Then any line that is perpendicular to this line would have a slope that is the negative reciprocal of 4, so we flip 4 to get 1/4 and then we switch the sign to get -1/4. Thus, any line perpendicular to a line with slope 4 has slope -1/4.

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