Lesson Video: Complementary and Supplementary Angles Mathematics • 7th Grade

In this video, we will learn how to identify complementary and supplementary angles and apply these relationships in order to find a missing angle.

08:03

Video Transcript

In this video, we’ll learn how to identify complementary and supplementary angles and apply these relationships to find missing angles. Firstly, let’s begin by reminding ourselves of two important angle facts. The first is that a right angle is 90 degrees. The second property is that angles on a straight line add up to 180 degrees.

So let’s imagine we have this 30-degree angle and this 60-degree angle. We can see that these two angles would clearly add up to give us a right angle of 90 degrees. We would say that these two angles are complementary. We say that two angles are complementary when they add up to 90 degrees.

You might think, “Actually, I’ve heard the word ’complimentary’ before, and it didn’t mean anything about adding up to 90 degrees.” Notice that this spelling of the word “complimentary,” where you say something nice about somebody else, is spelt slightly different with an “i” in the middle.

Let’s move on to having a look at supplementary angles. Here, we have a pair of angles of 50 degrees and 130 degrees. And these two angles would be supplementary, but why? It’s because they add up to 180 degrees. When it comes to both complementary and supplementary angles, the angles in question don’t need to share a vertex. For example, these two angles of 30 degrees and 60 degrees would still be complementary because they add up to 90 degrees. This 50-degree angle and this 130-degree angle would still be supplementary.

We’ll now have a look at some questions.

Classify the pair of angles as complementary, supplementary, or neither.

If we have a look at the diagram, there are two angles. One is 101 degrees, and the other is 79 degrees. So let’s begin by recalling our definitions of complementary and supplementary angles. Angles which are complementary are those which add to give 90 degrees, and those that are supplementary will add to give 180 degrees. When we add up the angles of 101 and 79 degrees, we can easily see that this would give us 180 degrees. We can, therefore, give our answer that these pair of angles would be supplementary.

In our next question, we’ll think about the angles in a right triangle.

Does every right-angled triangle contain a pair of complementary angles?

Let’s start by thinking about what a right-angled triangle or right triangle will look like. No matter the shape or size of the right triangle, we know that it would always have a right angle.

Next, we need to remember what complementary angles are. These are a pair of angles that add up to 90 degrees. So in the right triangles, do we have a pair of angles which add to 90 degrees? We know that one of the angles in the triangle is 90 degrees. But what about the other two?

We can remember that the angles in a triangle add up to 180 degrees. So if we take the other two angles and add them together, we must get 90 degrees because our right angle of 90 degrees plus the other two angles of 90 degrees would give us 180 degrees, which would be the sum of the three angles. Therefore, two of the angles or a pair of angles must be complementary. So we can give our answer to the question as yes, every right-angled triangle does contain a pair of complementary angles.

In the next question, we’ll find the missing angle in a pair of supplementary angles.

Given that the two angles are supplementary, find the value of 𝑥.

The diagram shows us an angle of 𝑥 and an angle of 89 degrees. Let’s begin by remembering that if we have two angles which we’re told are supplementary, it means that they add up to give us 180 degrees. This means that 𝑥 and 89 must sum to 180 degrees. So we could, therefore, work out the value of 𝑥 by calculating 180 degrees subtract 89 degrees. This would give us our answer that 𝑥 is equal to 91 degrees.

Let’s have a look at another question.

Given that the measure of angle 𝐴𝑂𝐵 equals 75 degrees, what is the measure of angle 𝐵𝑂𝐶?

Let’s start by filling in this angle information onto the diagram, that angle 𝐴𝑂𝐵 is 75 degrees. We need to find out the measure of angle 𝐵𝑂𝐶. We can work this out once we realize that this angle at 𝐴𝑂𝐶 is given as a right angle of 90 degrees. So our two angles 𝐴𝑂𝐵 and 𝐵𝑂𝐶 must add to give us 90 degrees. We could in fact say that these two angles are complementary. The size of angle 𝐵𝑂𝐶 could be calculated by working out 90 degrees subtract 75 degrees. And therefore, we can give our answer that the measure of angle 𝐵𝑂𝐶 is 15 degrees.

Let’s have a look at one final question.

In the given figure, Matthew stated that 𝑥 is an obtuse angle measuring 105 degrees, and Daniel stated that 𝑥 is an acute angle measuring 75 degrees. Determine which of the two is correct without using a protractor.

In the diagram, we’re given three angles: a 63-degree angle, a 42-degree angle, and 𝑥. We’re told here not to use a protractor, so we shouldn’t try and measure angle 𝑥 directly. When we see this type of instruction, especially in an exam question, very often the actual angle isn’t drawn accurately.

We’ll therefore need to find a way to work out the value of 𝑥 without measuring. That way, we’ll be able to tell whether Matthew is correct or Daniel is correct. You may have already noticed that these three angles sit upon a straight line. And we should remember that the angles on a straight line add up to 180 degrees. We can also say that these three angles are supplementary as supplementary angles add up to 180 degrees.

If we add the value of the two other angles, 63 degrees and 42 degrees, well, 60 and 40 would give us 100, and three and two would give us five. So we know that these two angles would add to 105 degrees. But we need to find the value of the angle 𝑥, which is remaining. Since these three angles add to 180 degrees, we would calculate 180 degrees subtract 105 degrees, which gives us the value of 75 degrees. As Daniel was the person who correctly identified that 𝑥 is 75 degrees, then Daniel would be our answer.

We’ll now summarize what we’ve learned in this video. Two angles are complementary if they add up to 90 degrees, and two angles are supplementary if they add up to 180 degrees. It’s worthwhile spending some time trying to learn these keywords as it’s very common to get the two definitions mixed up.

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