### 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.