# Parallelogram

A special type of **polygons** is known as a **parallelogram** . This is a quadrilateral where both pairs of opposite sides are **parallel** . The word has its origin in the Latin word **parallelogrammus** , and this concept helps us to identify a **quadrilateral** in which the opposite sides are parallel to each other. This geometric figure constitutes is then formed by a polygon that is made up of **4 sides** where there are two cases of **parallel sides** .

## What is a parallelogram?

It is a polygon that is formed by **four sides** and that is characterized because its **opposite sides** are always **parallel to** each other or, in other words, they are located at the same distance from each other.

- Characteristics of the parallelogram
- Properties
- Classification
- Elements
- Parallelogram law
- Height
- Diagonals
- Area
- Perimeter
- Angles
- Parallelogram method
- Examples

## Characteristics of the parallelogram

The main characteristics that we can observe in a parallelogram are the following:

- It will always have two pairs of
**sides**that are**parallel**. - Besides having a pair of sides that are
**parallel**they are also**equal**. - Opposite sides are
**equal to pairs**. - The diagonals intersect at one
**point**, in the middle or center of the parallelogram. - Their
**opposite sides**never meet. - The
**sum**of all**interior angles**will always be**360 degrees**. - Angles that are
**opposite**have equal measure. - All parallelograms are
**convex**. - Each parallelogram has four sides and an equal number of
**vertices**.

## Properties

The properties that characterize parallelograms are the following:

- The pairs of
**opposite sides**that the parallelogram has will always be**equal**. - The
**pairs of angles**that are opposite are**equal**. - Every two
**contiguous angles**are supplementary and add up to a total of 180. - Your two
**diagonals will**always intersect at their**midpoints**.

## Classification

It is important to know that **squares** , **rectangles** , **rhombuses, trapezoid, trapezoid, polygon, cube** and **rhomboids** are parallelograms, and their main characteristics are the following:

**Square**: its four**sides**are equal and its four**angles**are right.**Rectangle**: its four**angles**are right.**Rhombus**: all four**sides**are equal, but it has two different angles two by two, for this reason, the**adjacent**angles will be different and each of its angles is equal to the adjacent angle.**Rhomboid**: it has its four sides that are not equal and there is no right angle in them. It is also known as a**non-regular parallelogram**.**Cube**: it is a body made up of six faces and each of them is square.**Polygon**: it is a two-dimensional figure that has**straight lines**that connect in a closed way.**Trapezoid**: geometric figure with**four sides**that are not parallel.**Trapezoids**: a four-sided geometric figure in which two sides are**parallel**.

## Elements

Parallelograms have three different elements that make them up, these are:

**Sides**: they have**four sides**, being equal and parallel two by two (a and b).**Angles**: the interior angles that parallelograms have are**equal to**two by two, the non-consecutive**angles**(α and β) being equal .**Diagonals**: if the diagonals (D1 and D2) are**perpendicular**, the parallelogram will be a**square**or a**rhombus**. If the diagonals are**equal**, it is a**square**or a**rectangle**. These two diagonals can be calculated using the parallelogram law.

## Parallelogram law

There is a geometric law that aims to relate the sides of a parallelogram with its diagonals, this is known as the parallelogram law. The law tells us that the sum of the squares of the lengths of the four sides of any parallelogram will always be equal to the sum of the squares of the lengths of the two diagonals. It can be represented by the following formula:

**(AB) ^{2} + (BC) ^{2} + (CD) ^{2} + (DA) ^{2} = (AC) ^{2} + (BD) ^{2}**

In which A, B, C, and D are the vertices of the parallelogram.

## Height

Height is represented by the letter h and is calculated by dividing the area by the base of the parallelogram.

**h = A / b**

## Diagonals

A diagonal is a **segment** of **straight** connecting the **interior vertex** having a geometric shape with the **apex** located **opposite** and is not consecutive to it. In the parallelograms there is a **theorem** that says that if a quadrilateral is a parallelogram, then the **diagonals** are **bisect** one another and that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

## Area

The area of a parallelogram is the product of multiplying the **base times** the **height** . The base is any of its **sides** and the height is the **distance** between the **base** and its **parallel side** . The formula is as follows:

**A = b * h**

Where b is the base and h is the height.

## Perimeter

The Perimeter, represented by the letter **p** , can be calculated as the **sum** of its four **sides** . Thinking that its opposite sides are equal, we can indicate the perimeter with the following formula: **p = 2 a + 2 b**

Being a and b the length of two non-consecutive sides of the parallelogram, or taking a common factor we would have: **p = 2 (a + b)**

## Angles

The interior angles that a parallelogram has are **equivalent** to the **sum** of the **angles** of the two triangles that are inside. The sum of these interior angles must be **306 °.**

## Parallelogram method

This method is a very simple procedure that allows us to find the **sum** of two **vectors** . The first step is to draw both vectors, **a** and **b** to scale, with a common application **point** . The second step is to complete a parallelogram by drawing two **segments** that are **parallel** to them.

The vector sum that results from the operation a + b will be the **diagonal** of the parallelogram.

## Examples

Some examples of parallelograms are:

- Straight Trapezius
- Scalene Trapezoid
- Trapezium isosceles
- Squares
- Rectangle
- Diamond