What are Supplementary Angles?

Angles are everywhere, whether we see them or not. Supplementary angles are just one of many types of angles out there that hold up our world.

Unlike acute and obtuse angles, the term “supplementary” refers to not just one but several angles, all next to each other. For a set of angles to be supplementary, all angles must add up to 180 degrees. Supplementary angles can be made of both acute and obtuse angles. 

Supplementary Angles vs. Complementary Angles

While the sum of all supplementary angles must equal 180 degrees, the sum of all angles within a complementary angle must equal 90 degrees. Thus, two right angles can be supplementary to each other as long as the line separating them is perpendicular to a straight line.

Since supplementary angles can be composed of more than two angles, you could end up with complementary and supplementary angles in the same structure.

Examples of Supplementary Angles

The sum of all angles that make up a supplementary angle must equal 180 degrees. When calculating the size of an angle opposite another, use the formula [X + Y = 180]. 

Here are some examples of supplementary angles:

  • 90 + 90 = 180
  • 60 + 30 = 180
  • 75 + 105 = 180
  • 90 + 20 + 70 = 180
  • 45 + 35 + 28 = 180

Supplementary Angles in Real Life

In real life, supplementary angles are everywhere. Walls usually run perpendicular to floors, and chair or table legs sit at different angles against floors as well. Cutting something into circular slices creates many sets of supplementary angles. 

One of the best ways to spot a supplementary angle in real life is to look for scaffolding or bridges. They use tons of supplementary angles to create structural support.

Image source: https://unsplash.com/photos/EqUFBrTIdeE

Image description: A set of metal scaffolding with stairs. Notice how the supports form both right and supplementary angles against the metal. The stairs also create supplementary angles with the flooring of the scaffold.

Image source: https://unsplash.com/photos/XG8BPhYhLHc

Image description: An old railroad bridge. It uses several parallel beams to create a base and perpendicular ones to shape the skeleton. Beams set at various angles create triangles within the structure to keep it sturdy after many years. 

Example Problems:

Here are some examples of problems that involve supplementary angles. Feel free to try them at your leisure.

Question 1

Two lines meet a ray, creating an isosceles triangle. The two equal angles measure 40 degrees. How big is the third angle?

Answer: 100 degrees

Question 2

A signpost leans 5 degrees to the left when it should be perpendicular to the ground. What are the new measures of each angle?

Answer: 95 degrees, 85 degrees

Question 3

Throckmorton wants to cut his pizza into eight equal slices. He first makes one horizontal cut and then a vertical cut perpendicular to the first one. What should he do next?

Answer: He cuts a line that bisects the right angle.

Example Problems, Hard Mode:

Question 1

A four-way intersection consists of one road that is bisected by another. The upper left corner of the intersection is 76 degrees. Find the other degree measurements of each intersection.

Answer: 76 degrees, 104 degrees, 104 degrees

Question 2

The Department of Transportation needs to repair a stop sign that has been hit by a car. The sign is currently tilted 60 degrees backward. How far will the workers have to adjust the sign, so it is perpendicular to the ground?

Answer: 30 degrees

Question 3

Two parallel lines bisected by a ray create one angle that is 34 degrees. If you were to make a right triangle with lines perpendicular to the parallel ones, what would the last angle be?

Answer: 56 degrees