# Parallelogram Definition

## Introduction:

The discovery of mathematical figures is one of the greatest achievements of humanity. Due to the development of different types of figures, we can create 2D and 3D shapes. The development of figures is not only limited to mathematics, but it has wide applications. Today, these figures used in every field help create some good shapes. Everything we observed today is in the shape of different mathematical figures; for example, when we observe our homes, we find different types of shapes in different things. For instance, the TV has a rectangular shape; similarly, the bed is also in the shape of a rectangle. It means that in our daily life, we witness different figures. This article will discuss one of the most notable figures in mathematics: Parallelogram.

A parallelogram is a geometrical figure or shape with parallel sides. In other words, it is a shape with four sides and parallel opposite sides. It is a type of quadrilateral due to the presence of four sides and under which parallel sides are of equal length. One hundred eighty degrees is the sum of adjacent angles of the Parallelogram. When we talk about 2D shapes, we already know there are numerous types of 2D shapes; for example, circles, squares, rectangles, rhombus, etc., all are 2D shapes. Despite being 2D figures, they have different sets of properties, and because of that, they have different formulas to find the area or perimeter of the shapes.

Let us now learn here a parallelogram's definition, formulas, and properties.

## Parallelogram Definition:

Parallelogram is a type of Quadrilateral that have two sides in parallel form. The most striking feature of a parallelogram is that its opposite sides are equal in length. Also, opposite angles of the Parallelogram are equal in measure. The interior angles of the Parallelogram on the same side of the transversal are supplementary. Three hundred sixty degrees is the sum of all the interior angles of the Parallelogram. Parallelepiped is the shape that developed when we observed Parallelograms in 3D.

The area and perimeter of the Parallelogram depend upon the sides of the Parallelogram. For instance, its area depends upon the base and height of the figure, whereas the perimeter is the length of the four sides of the Parallelogram. Some figures in mathematics have similar properties with parallelograms, for instance, squares and rectangles. These two figures have similar properties to Parallelogram. Rhombus is another figure similar to a parallelogram; the main difference between a rhombus and Parallelogram is that a rhombus has all sides equal.

Another figure in a similar family is a trapezium; if one parallel side is parallel and the other two are non-parallel, then the figure is a trapezium.

Some of the most important facts of the Parallelogram are-

• Number of sides in parallelogram = 4
• Number of vertices present in parallelogram = 4
• Number of Mutually Parallel sides in parallelogram= 2 (in pair)
• Area of the parallelogram = Base x Height
• The perimeter of the parallelogram = 2 (Sum of adjacent sides length)
• Type of polygon that Parallelogram is = Quadrilateral

## Real-Life Examples of Parallelogram:

Everything around us is in the shape of anything. We found different types of shapes when we looked around. We observed different types of parallelogram shapes around us; for instance, we found parallelograms in buildings, paper, boxes, etc. Some of the most notable real-life examples of Parallelogram are-

### 1. Buildings:

In today's times, many buildings are developed in the shape of parallelograms. Due to advancements in technology and shape development, architecture developed hybrid models of the building by mixing different shapes, giving them unique shapes. When we observed around us, we witnessed numerous Parallelogram-shaped buildings. The most remarkable example of a parallelogram-shaped building is the Dockland Office Building in Hamburg, Germany.

### 2. Tiles:

When we construct our houses, we need several things. One of the most important things that increase the beauty of our home is the type of tiles used in construction. Today different types of tiles are used to increase the house's attractiveness. The most common tiles shape is a parallelogram. Parallelogram-shaped tiles are used in buildings, houses, and offices to increase the beauty of the place.

### 3. Eraser:

We use several things in daily life in the shape of a Parallelogram. For example, everyone is familiar with the use of erasers in school times. Erasers also come in several shapes and sizes, but the most common shape of an eraser is Parallelogram. The faces of this eraser are in the shape of a parallelogram.

In the above paragraphs, we discussed the definition of Parallelogram. Now we are going to discuss some of the special cases of Parallelogram. There are some figures which considered parallelograms because of some properties. Some of the special cases are-

1. Rectangle - A rectangle is a figure that is similar to a parallelogram because it has four angles of equal size, and the angles are right angles.
2. Rhombus - Rhombus is a figure with four sides of equal length and is also similar to a parallelogram. A Rhomboid is a figure that is neither a rectangle nor a rhombus; this figure is not used in current times.
3. Square - A square is a type of Parallelogram with four sides of equal length and angles of equal size. The angles formed by the square are right angles.

## Properties of Parallelogram:

Numerous figures are similar to parallelograms. To identify the Parallelogram, the best way is to find its properties. The properties of Parallelogram are very important because it helps to create shape perfectly. We already discussed the definition of Parallelogram and some of the special cases of Parallelogram, but what makes Parallelogram a true figure? Let us discuss this now. Firstly Parallelogram is a non-self-intersecting quadrilateral, which makes it different from the other figures. Let us now discuss some of the most remarkable properties of the Parallelogram-

1. The first and most important property of a parallelogram is that it has two pairs of parallel opposite sides. Both pairs are parallel to each other.
2. Second important property of the Parallelogram is that its two pairs of opposite sides are equal in length.
3. The opposite pair of angles formed by the Parallelogram are equal in measure.
4. The diagonals of the Parallelogram bisect each other.
5. The other remarkable property of the Parallelogram is that it's one pair of opposite sides is parallel and equal in length.
6. The Adjacent angles formed by Parallelogram are supplementary.
7. The diagonals formed by Parallelogram divide the Quadrilateral into two sets of congruent triangles.
8. According to parallelogram law, the sum of the squares of the sides of a Parallelogram is equal to the sum of the squares of the diagonals.
9. Parallelogram has rotational symmetry that has an order of 2.
10. There is an extension of Viviani's theorem that applies to the Parallelogram, which states that the sum of the distances from any interior point of the Parallelogram is independent of the location of the point.

Thus, above discussed property of Parallelogram is sufficient to identify any figure as Parallelogram. And if any figure satisfies any of the statements, it is a quadrilateral, and ultimately it is a parallelogram.

## Some Other Useful Properties of Parallelogram

1. We already know that opposite sides of a parallelogram are parallel, but these opposite sides never intersect each other.
2. Parallelogram forms triangles, but the area of a Parallelogram is twice the area of a triangle created by its diagonals.
3. The magnitude of the cross product of two adjacent sides of a parallelogram is equal to the area of a parallelogram.
4. When we draw any line through the midpoint of the Parallelogram, it bisects the area.
5. Parallelogram has rotational symmetry of order 2 when we look through 180°, or it has ordered four if a figure is square. Rhombus has two lines of reflection symmetry, whereas square has four lines of reflectional symmetry.

## Formulas Used in Parallelogram:

1. Area of Parallelogram:

The area of a parallelogram is defined as the region occupied by a parallelogram in a two-dimensional plane. The formula to find the area of Parallelogram is -

Area = Base × Height

2. Perimeter of Parallelogram:

The perimeter of the Parallelogram is the total distance covered by the boundaries of the Parallelogram. First, we have to find the length and breadth of a parallelogram to find its perimeter. And Parallelograms have opposite sides in equal length, so the formula to find the perimeter of the Parallelogram is-

Perimeter = 2 (a+b)

Where a and b are sides of the Parallelogram.

## The Conclusion:

Parallelogram is one of the essential figures in mathematics.

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