“Wow!” Juanita exclaimed looking at a glass structure outside the art museum. “Come look at this,” she called to her friend Samantha.

Samantha came over to see what Juanita was so excited about.

“What’s going on?” Samantha asked.

“Look at all of the figures in this sculpture,” Juanita commented. “You can see so many different ones. I can see squares, trapezoids and triangles.”

“Oh, yeah, I can see those too,” Samantha agreed.

**Do you? This Concept is about identifying quadrilaterals. At the end of the Concept you will be able to identify those figures in the glass structure.**

### Guidance

In this lesson we will examine figures called polygons. *Polygons***are closed shapes with sides made up of lines.** Any shape with straight edges, such as a triangle or rectangle, is a polygon. We will learn to identify each kind of polygon. Let’s take a look at some polygons.

**Polygons with four sides, such as rectangles and squares, are called** ** quadrilaterals.** Quadrilaterals have special properties.

**What is a quadrilateral?**

The prefix of this word “quad” means four. **Therefore, a quadrilateral is a polygon that has four sides. They also have four angles, and these four angles always have a sum of** \begin{align*}360^\circ\end{align*}. **This is true no matter what shape or size the quadrilateral is.** Take a look at the quadrilaterals below.

Each figure has four sides and four angles. Notice how different the angles and the sides of the quadrilaterals are, though. Now look closely. If you add up the measures of the four angles in any of the quadrilaterals, they always equal \begin{align*}360^\circ\end{align*}!

**Another important feature of quadrilaterals is that many have one or even two pairs of parallel sides.** Look at these quadrilaterals again.

**Can you find the pairs of parallel sides?**

**We can also classify quadrilaterals even further. You can see that each of the figures above has four sides and four angles, but each is different based on the lengths of the sides, the angle measures and the types of sides.**

**Let’s look at the different types of quadrilaterals in detail.**

**A** *parallelogram***is a quadrilateral with opposite sides parallel.** The first figure above is a parallelogram. There are a few special features of parallelograms. One is that each pair of parallel sides is congruent. In the first figure above, the two short sides are the same length and the two long sides are the same length. This special relationship between the pairs of sides affects the angles of a parallelogram. This gives parallelograms their other distinguishing feature: the angles opposite each other are also congruent. The \begin{align*}70^\circ\end{align*} angles are opposite each other, and the \begin{align*}110^\circ\end{align*} angles are opposite each other. This relationship exists in any parallelogram, no matter the length of the sides or the sizes of the angle pairs.

**There are three special kinds of parallelograms:** *rectangles, squares, and a rhombus.*

A ** rectangle** is a quadrilateral because it has four sides, and it is a parallelogram because it has two pairs of parallel, congruent sides. Now take a look at its angles. All four angles are right angles! Therefore a rectangle is any shape with two pairs of parallel sides and four right angles (bear in mind that the pairs of angles opposite each other are still equal!).

A ** square** also has two pairs of parallel sides and four right angles. It is special, though, because all four of its sides are congruent.

The third special parallelogram is called a ** rhombus.** Like a square, a rhombus has four congruent sides. It does not have right angles, but it does still have pairs of congruent angles opposite each other. So a rhombus can be a square, but a square isn’t necessarily a rhombus because of the right angles necessary in a square.

Let’s try recognizing and classifying some parallelograms.

Identify the shapes below as a rectangle, square, rhombus, or parallelogram.

We know that all parallelograms have two pairs of parallel sides. To distinguish them, we need to analyze the angles and compare the lengths of each pair of sides.

The first figure does not have right angles, so it cannot be a rectangle or square. Now compare the sides. One pair is 5 centimeters and the other pair is also 5 centimeters. **Four congruent sides without right angles make this a rhombus.**

Figure 2 does have right angles, so it must be either a rectangle or a square. Compare the pairs of sides to find out. One pair is 2 inches, but the other pair is only 1 inch. **This figure does not have four congruent sides, so it is a rectangle.**

Now let’s look at Figure 3. No right angles here. What about the sides? The pairs are not the same length, so it cannot be a rhombus. **This is a parallelogram; it does not have any other special features.**

**The next figure does. It has four right angles and four congruent sides, so it is a square.**

**Now let’s look at one other special quadrilateral. A** *trapezoid***is a quadrilateral that has only one pair of parallel sides.**

**Keep the characteristics of each type of quadrilateral in mind as you identify different four sided figures!**

Here is a chart with all that you need to identify the different quadrilaterals.

We’ve learned to identify all the special kinds of quadrilaterals. Remember also that no matter how long a quadrilateral’s sides are, or whether or not any sides are parallel, their four angles always add up to \begin{align*}360^\circ\end{align*}.

Identify each type of quadrilateral based on its description.

#### Example A

A four sided figure with opposite sides parallel.

**Solution: Parallelogram**

#### Example B

A four sided figure with opposite sides parallel and congruent with four right angles.

**Solution: Square**

#### Example C

A four sided figure with two pairs of parallel sides. Opposite sides are congruent.

**Solution: Rectangle**

Remember the glass structure? Well now that you have learned all about quadrilaterals look at the sculpture again.

Make a list of all of the quadrilaterals that you can see. Then share your list with a partner and compare answers. Did you find all the same ones? Add any other quadrilaterals to your own list.

### Vocabulary

- Polygon
- simple closed figure made up of straight lines and angles. Polygons are identified by the number of sides and angles in them.

- Quadrilateral
- a four sided figure.

- Parallelogram
- a quadrilateral with opposite sides parallel.

- Rectangle
- a parallelogram with opposite sides congruent, parallel and with four right angles.

- Square
- a rectangle with four congruent sides.

- Rhombus
- a parallelogram with four congruent sides.

### Guided Practice

Here is one for you to try on your own.

Name this figure.

**Answer**

This is a four sided figure with one pair of parallel sides.

**It is a trapezoid.**

### Video Review

James Sousa, Angle Relationships and Types of Triangles

### Practice

Directions: Identify each quadrilateral as a parallelogram, rectangle, square, rhombus, or trapezoid.

1.

2.

3.

4.

Directions: Look at each image and name the quadrilateral pictured.

5.

6.

7.

8.

9.

Directions: Name the geometric figure described below or answer each question.

10. Has four sides and four angles

11. Has one pair of opposite sides that are parallel

12. Has four right angles and four congruent sides

13. A parallelogram with four right angles.

14. A parallelogram with four congruent sides

15. Has four right angles and opposite sides congruent