Quadrilaterals are very important figures in geometry and you may have some questions about them: what quadrilaterals are, how they are classified, what examples of quadrilaterals there are, and how the different types of quadrilaterals are constructed.

All elementary and high school students should know about these four-sided polygons, and in this article we will help you learn everything you need to know about these varied and interesting geometric figures.

## What does the word quadrilateral mean?

Let’s start with the etymology of the word, that is, what quadrilateral means.

In Latin quadrilateral is written * chyadrilaterus*and the roots of this word are:

**Quadri**which means four in Latin, and

**lateris**which means side.

So quadrilateral (quadrilaterus) means four-sided figure.

## Elements of quadrilaterals

Quadrilaterals have, as we already said, four sides, but they also have other elements that compose and characterize them. Quadrilaterals have:

**Four vertices**.

**Two diagonals.**

**Four angles.**

An important property of quadrilaterals is that the measure of their internal angles adds up to 360°.

## How are quadrilaterals classified?

Quadrilaterals are classified according to the parallelism of their sides into three large groups:

**Parallelograms:** They are quadrilaterals that have their opposite sides parallel.

**Trapeze:** They are quadrilaterals with a single pair of opposite parallel sides.

**Trapezoids:** They are quadrilaterals without parallel sides.

Look at this diagram that explains the **classification of quadrilaterals**:

** **

### How are parallelograms classified?

As you can see in the diagram, parallelograms are classified into:

**Rectangles:** They are parallelograms with their **four right angles**.

**Rhombuses:** They are parallelograms with their **four equal sides**.

**Rhomboids:** They are parallelograms with unequal adjacent sides, two obtuse angles and two acute angles.

### Activities with quadrilaterals

Read each of these statements and decide if what it indicates is true or false:

** A trapezoid is a parallelogram.**

**Fake**. Trapezoids have only one pair of opposite sides parallel, while parallelograms have their two pairs of opposite sides parallel.

** A square is a trapezoid.**

**Fake**. Squares have four sides that are equal and parallel two to two. Trapezoids have a single pair of parallel sides.

** A rectangle is a parallelogram.**

**TRUE.** Rectangles have their opposite sides parallel, which is why they are parallelograms.

** A square is a rectangle.**

**TRUE.** One of the characteristics of squares is that their opposite sides are parallel, just like rectangles, and that all their angles are right.

** A parallelogram is a rhombus.**

**Fake.** Rhombuses are parallelograms with all equal sides, but not all parallelograms have equal sides. An example of this is the rectangle, whose equal sides are only opposite ones.

** A rhombus is a square.**

**Fake.** Although rhombuses and squares have four equal sides, only squares have all right angles.

** A square is a rhombus.**

**TRUE.** The definition of a rhombus is “a quadrilateral with four equal sides,” and that condition is met by the square.

**Some quadrilaterals are squares.**

**TRUE.** Among quadrilaterals there are squares, rectangles, trapezoids, trapezoids, and parallelograms.

** Some parallelograms are rhombuses.**

**TRUE.** Among the parallelograms we find rhombuses, also rectangles and rhomboids.

** Some quadrilaterals are parallelograms.**

**TRUE.** In addition to parallelograms, quadrilaterals are classified into trapezoids and trapezoids.

## What are squares?

Squares are quadrilaterals that have four equal sides and four right angles.

These two characteristics make them very special quadrilaterals because **They are rectangles and rhombuses at the same time.**.

Let’s look at different ways to represent the squares on the dotted grid:

As you can see, each of the squares has four equal sides and four right angles.

### Properties of squares

**Squares have four axes of symmetry**

First let us remember that symmetry is an exact correspondence in size, shape and position of the parts of an object. Furthermore, we will take into account that an axis of symmetry is a line that divides the figure into two identical parts.

We can easily visualize these axes of symmetry with a sheet of paper that has four equal sides, like the one you see below:

And you can make these folds to establish each axis:

**The diagonals of the squares intersect in the middle of the square**

A diagonal of a polygon is a line segment whose ends are two non-consecutive vertices of said polygon.

When drawing the diagonals of a square, they intersect at the midpoint of the square, and they also have the same length.

You can get the center of a square from a sheet of paper by making the folds we made to find the symmetry axis 3 and the symmetry axis 4.

### How to build a square?

To build a square we can follow these simple steps:

Given the **segment AB** We are going to draw a circle centered on A and with the opening of the compass measuring AB, that is, the radius is AB.

We now draw a segment perpendicular to AB that contains point A. And we point to point E, which is the intersection of the perpendicular that we have just drawn with the circle.

Now we will draw the line parallel to the segment AB that passes through point E.

We are now going to draw the perpendicular to the segment AB that contains point B, and we indicate point D, which is the intersection of this new perpendicular with the parallel to the segment AB that we drew in the previous step.

We can now draw the square ABDE

## What are rectangles?

Rectangles are parallelograms with four right angles.

Rectangles have different ways of being represented, let’s see some of them in the following dot plot.

Observe that in all the cases represented in the dotted plot they are parallelograms with their four right angles, that is, they are rectangles.

### Properties of rectangles

**Opposite sides have equal length.**

To verify this property we will take a rectangular sheet of paper and match its ends both horizontally and vertically.

When folding the sheet of paper, matching its ends, both horizontally and vertically, we realize that the opposite sides are of equal size. Also, it is said that the sides of a rectangle are equal two to two.

*In this way we verify this first property of rectangles.*

l**The diagonals have the same length.**

A rectangle has two diagonals of equal length. To visualize this property we will use a sheet of paper on which we will mark its diagonals. Then we will measure them with a graduated ruler.

Remember that a diagonal is a line segment whose ends are two non-consecutive vertices of the polygon.

*When measuring the diagonals with a graduated ruler we realize that both diagonals have the same measurement.*

**A rectangle has two axes of symmetry.**

First let’s remember that an axis of symmetry is a line that divides the figure into two equal parts in shape, size and position.

To visualize this property we will work again with a sheet of paper that we will fold in half both horizontally and vertically. Let’s see:

*The lines that we have highlighted in red are the axes of symmetry of the rectangle.*

### Building rectangles

In this section we will learn how to construct rectangles with a ruler and compass.

#### Construction of a rectangle given the measurements of its sides

In this case the base of the rectangle measures 6 cm and its height 4 cm.

We will begin by drawing the segment AB of length 6 cm, which is the base of the rectangle.

In this step we draw a perpendicular to segment AB from end A.

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Now take the compass and open it until you obtain an opening equal to the height of the rectangle, which in this case is equal to 4 cm. Then, centering on A, draw an arc of circumference that cuts the line perpendicular to the segment AB that passes through point A. This point is a vertex of the rectangle and we will call it D.

With the same opening that we used in step 3 and now centering on B we draw a circular arc. Let’s see:

We continue, now open your compass with a measure equal to that of segment AB. Center at D, draw an arc that cuts the arc drawn from B. The point of intersection of the arcs is the fourth vertex of the rectangle, we will identify it with the letter C.

We draw the segments BC and CD. In this way we obtain the rectangle ABCD.

## What are rhombuses?

Rhombuses are parallelograms with four equal sides.

#### Making rhombuses with ice cream popsicles

We are going to build a rhombus with ice cream popsicles.

**What do we need?**

4 Ice Cream Popsicles of equal size and 4 Thumbtacks.

**How do we build a rhombus with ice cream popsicles?**

Take two popsicles. Place one end of the paddle over the end of the other paddle. Do it in such a way that by placing one end over the other we can fit a tack so that the paddles are together. Repeat this process with the other two palettes. Then, join the two pairs of paddles together at the loose ends, placing a tack on each one. Look at the figure so you know what your rhombus should look like.

We have already built a physical model of the rhombus. Now you can move the paddles in such a way that you get different rhombuses with each movement. For example:

### Properties of rhombuses

**The diagonals of a rhombus are perpendicular.**

The first thing we will say is that a rhombus has two diagonals, the longest one is called **major diagonal** and the one with the shortest length is called **minor diagonal**. In the figure we have the rhombus ABCD, the major diagonal is the segment BD and the minor diagonal is the segment AC.

To verify that the diagonals AC and BD are perpendicular at point E, we measure the angle BEC with a protractor.

When measuring with the protractor, it is obtained that the angle BEC measures 90º, which *Verify that AC and BD are perpendicular.*

**Construction of a rhombus given the measurements of its diagonals**

Let’s see how we can build a rhombus in a few easy steps.

*We will build a rhombus, knowing the measurements of its diagonals, in this case the measurements are: 7 cm and 4 cm.*

We will draw the segment AB = 7 cm, which is the longest diagonal.

Now we will draw the bisector of segment AB. This construction will allow us to obtain a line perpendicular to segment AB at its midpoint.

Now on the bisector and centering on C, we mark with the help of a graduated ruler the measurement of the minor diagonal, which in this case is four centimeters. Since we must make the center at C, we mark 2 cm below C and 2 cm above C and in this way we obtain the minor diagonal of the rhombus, which is equal to 4 cm. We will name these points with the letters F and G.

Finally, we draw the segments AF, FB, BG, GA, obtaining the rhombus AGBF.