Have you ever examined street signs and their different shapes?

Jonas spent some time in the modern art section of the art museum. There was a fascinating exhibit on street signs. An artist had taken a bunch of signs and welded them together. The structure was ten feet tall.

Jonas noticed that many of the signs were stop signs.

A stop sign is an octagon. Is it a regular octagon or an irregular octagon?

**Use this Concept to figure out the answer to this question.**

### Guidance

**A polygon can have a certain number of sides, but the sides do not necessarily have to be the same length.** Both of the polygons below are pentagons because they both have five angles and sides. Look how different they are, though!

Count the angles or sides. Each has five, so each is definitely a pentagon.

**What makes them look so different?**

**Look at the lengths of the sides. In the first pentagon, all of the angles are congruent and all of the sides are congruent. In the second pentagon, the angles have different measures and the sides have different lengths.**

**We call a polygon whose angles and sides are all congruent a** ** regular polygon.** Any polygon can be a regular polygon.

Look at the regular polygons below.

As you can see, **a regular hexagon has six congruent sides.** It also has six congruent angles. Remember that the sum of a hexagon’s angles is always \begin{align*}720^\circ\end{align*}. Because the six angles in a regular hexagon are congruent, they are all @$\begin{align*}720 \div 6 = 120^\circ\end{align*}@$. **A regular hexagon, no matter how long its sides, will always have angles that measure** @$\begin{align*}120^\circ\end{align*}@$.

**A regular octagon has eight congruent angles and sides.** It doesn’t matter how long the sides, are as long as they are congruent. But every regular octagon does have the same angles. Can you find the measure of a regular octagon’s eight angles?

Well, we know that the sum of the interior angles in an octagon is @$\begin{align*}1,080^\circ\end{align*}@$. Because the angles are congruent, we divide by 8 to find the measure of each:

@$\begin{align*}1,080 \div 8 = 135^\circ\end{align*}@$

**Every regular octagon has angles of** @$\begin{align*}135^\circ\end{align*}@$.

We don’t usually say “regular rectangle,” because a rectangle with congruent sides is actually a square. As we know, the four angles in a square are always @$\begin{align*}90^\circ\end{align*}@$, and their sum is always @$\begin{align*}360^\circ\end{align*}@$. We also have a special name for a “regular triangle.” A triangle with three congruent sides is called an equilateral triangle. Its three angles are always @$\begin{align*}60^\circ\end{align*}@$, and their sum is always @$\begin{align*}180^\circ\end{align*}@$.

**Any polygon that does not have all congruent sides is an** ** irregular polygon.** Irregular polygons can still be pentagons, hexagons, and nonagons, but they do not have congruent angles or equal sides. Here are some examples of irregular polygons.

In these polygons, some sides are clearly longer than others. Some angles are wider than others. They cannot be regular polygons if their angles are different measures or their sides are different lengths, so they must be irregular. Notice that irregular polygons tend to look uneven or lopsided, while regular polygons look more orderly and symmetrical.

Let’s practice identifying irregular and regular polygons.

Identify each polygon below and tell whether it is regular or irregular.

**First, count the number of angles or sides to find which kind of polygon it is.**

Figure 1 has six angles and sides, so it is a hexagon. Do all the sides appear to be the same length? If you’re not sure, use a ruler to check. They are all equal, so this is a regular hexagon.

Figure 2 has seven angles and sides, so it is a heptagon. Notice how short the top side is compared to the bottom side. These sides are definitely not the same length, so this cannot be a regular polygon. Therefore it is an irregular heptagon.

The next figure has five angles and sides, so it is a pentagon. The angles are all @$\begin{align*}108^\circ\end{align*}@$, so this is a regular pentagon because it has congruent angles.

How many sides does this polygon have? It has eight, which makes it an octagon. It looks a bit lopsided, so it’s likely that the sides are not all congruent. Let’s take a look at the angles. Some of them are the same measure, but not all of them. It cannot be a regular polygon because all of its angles are not congruent; it is an irregular octagon.

**We can identify a polygon by the number of sides and angles it has, or by the sum of its interior angles. Then we can tell whether it is regular or irregular depending on whether all the angles or sides are congruent.**

**Now that you know about polygons, we can use this information to analyze polygons and their relationships.**

**When lines intersect in the geometric plane, they form polygons. We can apply what we know to these polygons in order to classify them, find side lengths, or solve for unknown angle measures. Take a look at the diagram below.**

**The intersection of these five lines has created several different polygons. First, let’s see if we can find them all.**

The largest is @$\begin{align*}AMDE\end{align*}@$. Within this figure there are two figures. What are they?

One is figure @$\begin{align*}ABCDE\end{align*}@$. What can we tell about figure @$\begin{align*}ABCDE\end{align*}@$? First, we can determine what kind of polygon it is. It has five angles and five sides, so it is a pentagon. Two of its angles measure @$\begin{align*}90^\circ\end{align*}@$, and one measures @$\begin{align*}120^\circ\end{align*}@$. Can we find the measures of the two remaining angles? We can, but to do so, we need to take a look at the third figure.

The third figure is triangle @$\begin{align*}MBC\end{align*}@$. It has two @$\begin{align*}60^\circ\end{align*}@$ angles. We can use these angles to help us find the measures of the unknown angles in the pentagon @$\begin{align*}ABCDE\end{align*}@$. Angles @$\begin{align*}ABC\end{align*}@$ and @$\begin{align*}MBC\end{align*}@$ are supplementary. In other words, together they form a straight line. A straight line measures @$\begin{align*}180^\circ\end{align*}@$. Therefore the sum of these two angles is @$\begin{align*}180^\circ\end{align*}@$. We simply subtract to find the measure of @$\begin{align*}ABC\end{align*}@$.

@$\begin{align*}180 - 60 = 120^\circ\end{align*}@$

Angle @$\begin{align*}ABC\end{align*}@$ is @$\begin{align*}120^\circ\end{align*}@$. Draw a copy of the diagram and fill this information in. Now let’s see if we can find the measure of angle @$\begin{align*}BCD\end{align*}@$. We now know four of the five angles in figure @$\begin{align*}ABCDE\end{align*}@$. Because this is a pentagon, we also know that its interior angles must have a sum of @$\begin{align*}540^\circ\end{align*}@$. We can set up an equation to find the measure of the unknown angle.

@$$\begin{align*}90 + 90 + 120 + 120 + \angle BCD &= 540^\circ\\ 420 + \angle BCD &= 540^\circ\\ \angle BCD &= 540 - 420\\ \angle BCD &= 120^\circ\end{align*}@$$

**The fifth and final angle must measure** @$\begin{align*}120^\circ\end{align*}@$. **Let’s add up all of the angles in the pentagon to be sure they total** @$\begin{align*}540^\circ\end{align*}@$.

@$\begin{align*}90 + 90 + 120 + 120 + 120 = 540^\circ\end{align*}@$

We still have one unknown angle, angle @$\begin{align*}AMD\end{align*}@$. Can we find its measure? In fact, we have two different ways! It is the third angle in triangle @$\begin{align*}BMD\end{align*}@$, and we know that the three angles in a triangle have a sum of @$\begin{align*}180^\circ\end{align*}@$. It is also the fourth angle in quadrilateral @$\begin{align*}AMDE\end{align*}@$, and we know that the four angles in a quadrilateral have a sum of @$\begin{align*}360^\circ\end{align*}@$. We can set up an equation to find @$\begin{align*}AMD\end{align*}@$.

@$$\begin{align*}60 + 60 + \angle AMD &= 180^\circ\\ 120 + \angle AMD &= 180^\circ\\ \angle AMD &= 180 - 120\\ \angle AMD &= 60^\circ\end{align*}@$$

We have used the properties of polygons to find all of the unknown angles in the diagram. Now that we know all of the angles, let’s classify each polygon as regular or irregular.

#### Example A

**Solution: Regular Polygon**

#### Example B

**Solution: Irregular Polygon**

#### Example C

**Solution: Regular Polygon**

Here is the original problem once again.

Jonas spent some time in the modern art section of the art museum. There was a fascinating exhibit on street signs. An artist had taken a bunch of signs and welded them together. The structure was ten feet tall.

Jonas noticed that many of the signs were stop signs.

A stop sign is an octagon. Is it a regular octagon or an irregular octagon?

A stop sign is an octagon, and all of the sides are the same lengths. This means that the octagon is a regular octagon because all of the side lengths are the same.

### Guided Practice

Here is one for you to try on your own.

Name this figure.

**Answer**

This is an irregular pentagon. This is a five-sided figure, but a few of the sides are different lengths.

### Explore More

Directions: Identify each quadrilateral by its description.

1. A quadrilateral with one pair of parallel sides.

2. A quadrilateral with opposites sides congruent and parallel, with four right angles.

3. A quadrilateral with four congruent parallel sides and four right angles.

4. A rectangle with four congruent parallel sides.

Directions: Identify the polygons in the diagram. Then find the measures of the unknown angles.

5.

Directions: Answer true or false for each of the following questions.

6. A rhombus is always a square.

7. A parallelogram has opposite sides that are parallel.

8. A rectangle is a type of parallelogram.

9. Squares, rectangles and rhombi are parallelograms with four right angles.

10. A trapezoid has four right angles.

11. A trapezoid has one pair of parallel sides.

12. A regular polygon has congruent sides and angles.

13. A pentagon can not be an irregular polygon.

Directions: Identify each polygon below.

14. a

15. b

16. c

17. d