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# Parallelogram|Definition & Meaning

## Definition

A **quadrilateral** that has **2 pairs of sides** that are **parallel** to one another is called a **parallelogram.** The lengths of the different sides of a parallelogram are equivalent to one another, and the opposite angles have the same degree of measurement. Additionally, the interior angles located on the same plane of the transversal are considered **supplementary.** The whole amount of interior angles add up to a total of 360 degrees.

**What Is a Parallelogram? **

The English word **“parallelogram”** was taken from this Greek word. As a result, a parallelogram is defined as a quadrilateral with **parallel lines** serving as its boundaries. It is a shape that has opposite sides that are parallel to one another and are of **equal length**.

Each of the three primary forms of parallelogramsâ€”square, rectangle, and rhombusâ€”is distinguished by a distinct set of characteristics distinct from the others.

In this article, we will study what a parallelogram is, how to calculate the area of a parallelogram, as well as other topics connected to parallelograms, followed by several instances that have been solved.

**Shapes of a Parallelogram**

A parallelogram is a shape that only exists in **two dimensions**. It possesses a total of **four sides**, two pairs of which are parallel to one another. Additionally, the** lengths of the sides are identical.** The form in question is not a parallelogram if the lengths of the parallel sides do not measure the same. Similarly, the parallelogram’s inner angles opposite one another should always be **equivalent**. In such a case, we cannot call it a parallelogram.

**Special Parallelograms**

Both square and rectangular: Two more shapes comparable to a parallelogram in terms of their qualities are a **square and a rectangle**. Both opposing sides are of the same length and run parallel. Both shapes can be cut in half along their respective **diagonals.**

**Rhombus**

A **rhombus** is defined as a parallelogram in which all of the sides are **congruent** with one another or are **equal.**

**Rhomboid**

A** rhomboid** is a specific example of a parallelogram in which the sides that are **perpendicular **to each other are also parallel to the sides that are **adjacent **to them, but the lengths of the sides that are perpendicular to each other differ. In addition, each angle adds up to precisely **90 degrees.**

**Trapezium**

A shape is said to be **a trapezium** if it has **one pair** of sides **parallel **to one another and the **other two sides** are** not parallel** to each other.

**Angular Measurements of a Parallelogram**

A parallelogram is a type of **two-dimensional** form that is **flat **and has **four angles**. Equal treatment is accorded to the opposing inner angles. The angles that are on the identical side of the transversal are **complementary**, indicating that their sum is equal to **180** **degrees**. Therefore, the total amount of angles that are contained within a parallelogram is equal to **360 degrees.**

**Parallelogram’s Characteristics and Attributes**

- A
**unique kind of polygon**known as a parallelogram is formed when a quadrilateral has two parallel sides and faces in opposite directions. The following is a list of the characteristics of a parallelogram: - The different sides are
**parallel**and**equivalent** - Equal to each other is the
**diagonal** - The angles that are
**successive**or**adjacent**to one another are**complementary**. - In the event that any of the angles is a right angle, then the remaining angles will also be
**right****angles**. - Both diagonals meet and
**cut**each other in**half**. - Each diagonal cuts the parallelogram in half, creating
**two congruent triangles**. - A parallelogram has a
**unique property**in which the sum of the squares of all sides is equivalent to the sum of the squares of its diagonals.

**Parallelogram’s Area**

The space enclosed by all four parallelogram’s sides is referred to as the **parallelogram’s area**. It is possible to calculate, given the length of the parallelogram’s base and the height of the parallelogram.

Take into consideration the **parallelogram RSTU** with the base symbol **(b)** and the height symbol **(h).** It is possible to get the area of the parallelogram by applying the following formula: The area of a parallelogram is equivalent to **the base times the height** (h)

**Parallelogram’s Perimeter**

The length of the shape of a parallelogram is its **perimeter**; hence, the perimeter is equal to the total of all of the parallelogram’s sides.

Therefore, the formula for calculating the perimeter of a parallelogram with sides is:

** P = 2 (a + b) units**

## Example Problems Involving Parallelograms

**Example 1**

A parallelogram has two adjacent angles of 4:5. Find each of its angles measurement.

**Solution**

Let the parallelogram be EFGH.

Then âˆ E and âˆ F are the adjacent angles.

Let:

âˆ E = (4x) **Â° **and âˆ F = (5x) Â°

We know that the sum of the adjacent angles of a parallelogram is equal to 180Â°, so:

âˆ E + âˆ F = 180Â°

4x + 5x = 180

9x = 180

x = 20

Now:

âˆ E = (4*20) Â° = 80Â°

âˆ F = (5*20) Â° = 100Â°

âˆ F + âˆ G = 180Â° (as âˆ F and âˆ G are adjacent angles)

100 + âˆ G = 180Â°

âˆ G = 180Â° – 100Â°

âˆ G = 80Â°

Also:

âˆ G + âˆ H = 180Â°

80Â° + âˆ H = 180Â°

âˆ H = 180Â° – 80Â°

âˆ H = 100Â°

Therefore, âˆ E = 80Â°, âˆ F = 100Â°, âˆ G = 80Â°, âˆ H = 100Â°

**Example 2 **

Consider a parallelogram ABCD in which âˆ B = 45Â°. Find the measure of each of the angles âˆ A, âˆ C, and âˆ D.

**Solution**

In the given parallelogram ABCD, âˆ B = 45Â°.

The total sum of any two adjacent angles is 180Â°, so:

âˆ A + âˆ B = 180Â°

âˆ A + 45Â° = 180Â°

âˆ A = (180Â° – 45Â°) = 135Â°

Also, we know that:

âˆ B + âˆ C = 180Â°

45Â° + âˆ C = 180Â°

âˆ C = (180Â° – 45Â°) = 135Â°

Furthermore:

âˆ C + âˆ D = 180Â° [since âˆ C and âˆ D are adjacent angles]

135Â° + âˆ D = 180Â°

âˆ D = (180Â° – 135Â°) = 45Â°

So, âˆ A = 135Â°, âˆ C = 135Â° and âˆ D = 45Â°.

*All images/graphs are created using GeoGebra.*