### Video Transcript

In this video, we’ll learn how to
identify exterior angles of polygons, find their sum, and use these to solve
problems.

We begin by defining the exterior
angle of a polygon. It’s an angle at a vertex of the
polygon outside the polygon, which is formed by one side and the extension of an
adjacent side. We see from our diagram the
exterior angle marked. Since these angles are made by
extending a side and angles on a straight line add up to 180 degrees, we see that
the corresponding interior and exterior angles are supplementary. That is, they add to 180
degrees. So, one way that we can calculate a
single exterior angle in a polygon is to subtract the measure of the interior angle
from 180 degrees. That isn’t always appropriate,
though. So, let’s look at a way in which we
can find the sum of the exterior angles in any polygon.

Olivia is experimenting with the
exterior angles of a triangle. She colors the angles, cuts them
out, and sticks them together as seen in the figure. Will angle 𝑧 fit in the space to
complete the circle?

One way we can answer this question
is to simply imagine the triangle getting smaller and smaller and smaller. We notice that as the triangle
shrinks to nothing, these lines appear to circle around a point. We know that angles around a point
sum to 360 degrees, so we can infer that angles 𝑥, 𝑦, and 𝑧 fit perfectly to
complete the circle. And so, we could add in angles 𝑧
on our original diagram as shown.

But is there a better way to prove
this? Well, yes, there is. We know the angles on a straight
line are supplementary. They add to 180 degrees. We also know the same about
interior angles in a triangle; they add up to 180 degrees. So, we’re going to fill in the
missing three angles in our first triangle. This first one must be 180 minus 𝑥
degrees. This must be 180 minus 𝑦. And over here, we have 180 minus 𝑧
degrees. We know that the sum of these three
interior angles is 180 degrees, so we can say 180 minus 𝑥 plus 180 minus 𝑦 plus
180 minus 𝑧 equals 180. This simplifies to 540 minus 𝑥
minus 𝑦 minus 𝑧 equals 180.

We’re going to subtract 180 from
both sides so that 360 minus 𝑥 minus 𝑦 minus 𝑧 equals zero. And then we’re going to add 𝑥, 𝑦,
and 𝑧 to both sides so that 360 equals 𝑥 plus 𝑦 plus 𝑧. We’ve actually proven that the
exterior angles 𝑥, 𝑦, and 𝑧 do indeed add to 360 degrees. And so, angle 𝑧 will fit in the
space and form a perfect circle. And so, the answer is yes. Note that we can generalize this
and say that the exterior angles of any polygon sum to 360 degrees no matter the
number of sides it has.

So, let’s look at how we can use
this formula to find an exterior angle of a regular polygon.

What is the measure of the exterior
angle of an equilateral triangle?

Let’s begin by recalling what we
know about the exterior angles of a polygon. We know that the exterior angles of
any polygon sum to 360 degrees no matter the number of sides it has. But we also know that all angles
and sides in an equilateral triangle are equal.

Essentially, it’s a regular
polygon. We can therefore assume that the
exterior angles of our equilateral triangle must all be equal in size. There are three of them. So, to find the measure of one
exterior angle of an equilateral triangle, we divide the angles’ sum, that’s 360
degrees, by three. 360 divided by three is 120. So, we can say that the measure of
the exterior angle of an equilateral triangle is 120 degrees.

Now, we can generalize this process
for any regular polygon. We can say that the measure of a
single exterior angle of a regular 𝑛-sided polygon, that’s a polygon with 𝑛 sides,
is 360 divided by 𝑛 degrees.

We’re now going to look at how we
can find the number of sides of a polygon given the measure of a single exterior
angle.

The measure of an exterior angle in
a regular polygon is 40 degrees. How many sides does the polygon
have?

Here, we have some information
about the size of one exterior angle in a regular polygon. Remember, a regular polygon is one
whose sides and angles are all equal. So, we recall that the measure of a
single exterior angle in a regular 𝑛-sided polygon is 360 divided by 𝑛
degrees. And that’s because the total angle
sum of the exterior angles is 360. Now, we don’t know how many sides
our polygon has, so 360 divided by 𝑛 here gives us the measure of a single exterior
angle, gives us 40 degrees. So, we can form an equation and say
that 360 divided by 𝑛 equals 40. Let’s look to solve this
equation.

We’ll begin by multiplying both
sides by 𝑛. 360 divided by 𝑛 times 𝑛 is just
360, and 40 times 𝑛 is 40𝑛. So, our equation is 360 equals
40𝑛. We’re looking to solve for 𝑛. Remember, we’re trying to find the
number of sides this polygon has. So next, we divide through by
40. So, 360 divided by 40 is equal to
𝑛. Well, 360 divided by 40 is
nine. And so, this regular polygon must
have nine sides.

We’ll now look at how we can use
the fact that exterior angles of a polygon add to 360 degrees to solve problems
involving irregular polygons.

The diagram shows an irregular
quadrilateral. Calculate the value of 𝑥.

If we look carefully, we see that
𝑥 is the measure of an exterior angle. That’s an angle on the outside of a
polygon formed by one side and the extension of an adjacent side as shown. In fact, we’re given two more
exterior angles and then the measure of a third angle. But this is the interior angle;
it’s inside the shape. And so, there are two ways we can
answer this question. Let’s begin by recalling what we
know about the exterior angles in a polygon. We know that the exterior angles in
a polygon sum to 360 degrees. We also know that since the
interior and exterior angles sit on single straight lines, a corresponding interior
and exterior angle sums to 180 degrees.

And so, let’s begin by working out
the exterior angle. Let’s call that 𝑦. We know that the sum of this and
its corresponding interior angle is 180 degrees. So we can say that 𝑦 plus 110 is
equal to 180. We’re going to subtract 110 from
both sides of this equation. And when we do, we find that 𝑦 is
equal to 70. And we now have three out of four
of our exterior angles in our irregular quadrilateral. We know that the sum of all four
must be 360 degrees. So, we can form a second equation,
this time 70 plus 43 plus 92 plus 𝑥, that’s all our exterior angles, equals
360. 70 plus 43 plus 92 is 205. And so, we solve for 𝑥 by
subtracting 205 from both sides. 360 minus 205 is 155. 𝑥 is equal to 155 or 155
degrees.

Now, in fact, there was a second
way we could’ve answered this, but it’s a slightly more long-winded method. In this method, we begin by
subtracting 43 and 92 individually from 180. In doing so, we calculate two more
of the interior angles. Then, since the interior angles in
a quadrilateral add to 360 degrees, we subtract the sum of 110, 137, and 88 from 360
to get 25. Then, 𝑥 is found by subtracting 25
from 180. Once again, that gives us 155 or
155 degrees.

Next, we’ll look at how we use the
sum of the exterior angles of a polygon to find an unknown.

The diagram shows an irregular
pentagon. Calculate the value of 𝑥.

Note here that this diagram is not
drawn to scale. In our pentagon, we’ve been given
one, two, three, four, five exterior angles. Remember, the exterior angles are
angles at the vertex but on the outside of the shape, made between one side and an
extension of the adjacent side. So, let’s recall what we actually
know about exterior angles of polygons. No matter its number of sides, the
exterior angles in a polygon all add to 360 degrees. So, let’s use this fact to form an
equation.

We’ll add together the given angles
from our diagram. That’s three 𝑥 plus three 𝑥 plus
two 𝑥 plus two 𝑥 plus two 𝑥. And we know that adds to make 360
degrees. We simplify by collecting like
terms, and our equation becomes 12𝑥 equals 360. Finally, we solve for 𝑥 by doing
the inverse operation. At the moment, it’s being
multiplied by 12. So, we’re going to divide both
sides of our equation by 12. 12𝑥 divided by 12 is 𝑥. So, we need to work out 360 divided
by 12. Well, 36 divided by 12 is three, so
360 divided by 12 is 30. And this means 𝑥 is equal to
30.

Now, where possible, we should try
and check our solutions. Here, we can check by substituting
our value for 𝑥 back into the original diagram. This angle will be three times 30,
which is 90 degrees. Similarly, we have 90 degrees
here. This angle will be two times 30
since it’s two 𝑥. That’s 60 degrees. And the remaining two angles are
also two 𝑥, so they’re 60 degrees. The sum of these five values should
be 360 degrees. So, let’s check. That’s 90 plus 90 plus 60 plus 60
plus 60, which is indeed equal to 360 as required. So, 𝑥 is equal to 30.

In this video, we learned that the
exterior angle of a polygon is the angle at the vertex of that polygon on the
outside of the shape. We learned that the sum of the
exterior angles of a polygon is 360 degrees. And if we have a regular polygon,
we can find the measure of a single exterior angle if it’s got 𝑛 sides by using the
formula 360 divided by 𝑛. Finally, we saw that each pair of
interior and exterior angles in a polygon are supplementary; they add to 180
degrees.