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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.

Examples of regular and irregular polygons, with equal sides and angles marked

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.

A chart detailing polygon names, their number of sides, and the measure of their interior angles

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.

An illustration of convex and concave polygons

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°.

A four-sided polygon to illustrate the polygon interior angle sum theorem

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.

A straight line drawn between two vertices on a polygon

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.

A chart detailing polygon names, number of sides, interior angle measures, and the number of triangles that can be formed

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.

A sample problem finding unknown angles in a four-sided polygon

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.

A sample problem finding an unknown angle using the interior angle sum theorem

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.

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