Polygons

Before we get too caught up on the excitement of
quadrilaterals, let’s take time to learn the names and basic properties
of different polygons. Polygons are two-dimensional, closed, plane shapes
composed of a finite number of straight sides that meet at points called vertices.
Triangles, which
we have already learned about, are one type of polygon. In this section we will
begin studying four-sided polygons, called quadrilaterals. There are many
more polygons than just triangles and quadrilaterals, however. In this section,
we will also learn the names and properties of some of the most common polygons to help
us get started with our study of quadrilaterals.

There are several ways to classify polygons. One way to classify them is by considering
their angle measures and side length measures. If a polygon’s angles and sides are
equal, then the polygon is called a regular polygon. If the measures of a
polygon’s angles or side lengths differ, then the polygon is called an irregular
polygon.

The shape on the left has equal sides and angles, so it is a regular polygon. Since
the shape on the right has side lengths and angles that differ, it is irregular.

Naming Polygons

There are a countless number of polygons. Because they all differ in the number
of sides that they have, this results in different angle measures at their vertices.
Listed below are the names and number of sides of some polygons. The “Interior Angle
Measure” column of the table only applies to regular polygons, in which
all the interior angles are equal.

With the exception of the triangle and quadrilateral, notice that all polygon names
end with “gon.” What sets regular polygon names apart from each other are their
prefixes, which speak to the number of sides that they have. For instance, the prefix
for the word “hexagon” is “hexa,” which essentially means “six.” However, as we
move down our list and the names for polygons becomes quite confusing, we need a
more efficient way of naming polygons. One way is by not calling a polygon by its
real name, but rather by just saying the number of sides it has, and attaching “-gon”
at the end. For instance, rather than calling an 18-sided polygon an “octdecagon,”
we can just call it an 18-gon. Thus, a polygon with n sides is simply
called an n-gon.

We will not study all of the polygons listed in the table, but they do share certain
properties worth looking at. Let’s examine one of these properties now.

Polygon Interior Angle Sum Theorem

The sum of the interior angle measures of an n-sided, convex polygon is

As seen in the statement above, the polygon must be convex, which is not
a term we have studied yet. What this means is just that the polygon cannot have
angles that point in. If a polygon does have an angle that points in, it is called
concave, and this theorem does not apply. In other words, all of the interior
angles of the polygon must have a measure of no more than 180° for
this theorem to work.

The Polygon Interior Angle Sum Theorem would apply for the polygon on the left (since
it is a convex polygon), but not for the one on the right because the highlighted
angle points in. That angle has a measure greater than 180°.

Now, let’s make sure that this theorem holds true for a polygon we have worked extensively
with: triangles. Recall that by the
Triangle Angle Sum Theorem, the interior angles of our triangle should measure
out to 180°. Let’s check this by using the Polygon Interior Angle Sum
Theorem.

Since triangles have three sides, we know that n=3. So let’s plug
this into our equation.

We see that the Polygon Interior Angle Sum Theorem is consistent with the triangle
theorem we have already studied.

Now, let’s figure out what the sum of the interior angles of a quadrilateral is.
Quadrilaterals have four sides, which means that n=4 for quadrilaterals.
Let’s plug this into our equation.

By this theorem, we see that all quadrilaterals have an interior angle sum of 360°.

Breaking Down the Polygon Interior Angle Sum Theorem

You may be asking yourself why the Polygon Interior Angle Sum Theorem works,
or if we can use it with any n-gon. Let’s investigate what the equation
means in order to assure ourselves that it works.

Essentially, what we do when we use the equation is split up the polygons into triangles,
and then we multiply by 180 because that is the sum of the angles
of a triangle. There is a two-step process we must follow in order to split up the
polygon into triangles, however. This process is listed below:

(1) Choose a vertex from which to draw straight lines from, and then

(2) draw a straight line from the chosen vertex to the other vertices of
the polygon (that are not already connected to the chosen vertex by a line segment).

Let’s try this out with the quadrilateral below since we already know that its interior
angles will measure out to 360°.

Let’s pick point A to draw lines from. Since there are already line
segments that connect point A to points B and D,
we only need to draw a line to point C.

Notice that we have created two triangles. Keeping the Triangle Angle Sum Theorem
in mind, we know that each triangle should have angles whose sum is 180°.
Thus, we multiply this measure by two (since there are two triangles), and indeed,
the sum of the angles of the quadrilateral is 360°.

For whatever polygon we have, we will always be able to create two less triangles
than the number of sides there are. Therefore, for any n-gon, we can
create (n-2) triangles from it. We multiply by 180 to
since that is the sum of the measures of a triangle. An extensive table of different
polygons and the sum of their interior angles is shown below.

Properties of Quadrilaterals

Let’s try a few exercises that use the property that the interior angles of quadrilaterals
add up to 360°.

Exercise 1

Determine the value of x in the figure below.

Solution:

We know that the sum of the interior angles of quadrilateral JESK
is 360°, so we have

Substitution with the angle measures we were given yields

Now, we just simplify our equation and solve for x.

So, we get x=9.

Exercise 2

Find the value of z in the figure below.

Solution:

Let’s begin this exercise by figuring out the value of the variables that are solvable
with the information that has been given. It appears as though we will not be able
to determine the value of z until we figure out the values of w,
x, and y.

We determine that w is equal to 78° because it is a
vertical angle to the 78° angle that has been provided.

Next, we can determine the value of x, since it is supplementary to
the 64° angle. We have

So we have figured out that x=116.

Now, we have three out of four angles of quadrilateral KLNM. Our new
figure (with the included degree measures of the variables we solved for) is shown
below.

We can solve for y since we know that the interior angles of the quadrilateral
must add up to 360°. We have

So, the value of y is 84.

Because that angle is a vertical angle to z, we conclude that the
measure of z must also be 84.