Properties of a Parallelogram: A parallelogram is a geometrical form with sides that are parallel to each other in two dimensions. It’s a four-sided polygon (sometimes known as a quadrilateral) with two parallel sides that have the same length. A parallelogram’s sum of neighbouring angles equals 180 degrees. Many 2D forms and sizes, such as circle, square, rectangle, rhombus, and so on, must have been covered in geometry class. Each of these forms has its own set of characteristics. These forms’ area and perimeter formulae differ from one another and are used to tackle a variety of issues. Let’s look at a parallelogram’s definition, formulae, and characteristics.

What is Parallelogram?

A parallelogram is a quadrilateral (four-sided plane figure) whose opposite sides are equal and parallel. The Sum of interior angles of a parallelogram is \(360°\) and the sum of each pair of adjacent angles of a parallelogram is \(180°\).

Properties of a Parallelogram Definition:

The property of a parallelogram can be defined as the collection of the different rules or relations by which the different parameters like sides, angles, vertices, diagonals etc. of a parallelogram are governed.

What are the Properties of a Parallelogram?

1. A quadrilateral is called a parallelogram if both pairs of its opposite sides are parallel and equal. Hence, \(AB=DC\) & \(BC=AD\).
2. The diagonals of the parallelogram bisect each other. Diagonals \(AC\) and \(BD\) bisects at \(O\).
3. The opposite angles are equal in a parallelogram. Here, \(∠A=∠C\) & \(∠B=∠D\).
4. The pair of adjacent angles are supplementary. Here, \(∠A+∠B=∠B+∠C=∠C+∠D=∠D+∠A=180°\).
5. The opposite sides are parallel and congruent.
6. The consecutive angles are supplementary.
7. If any one of the angles is a right angle, then all the other angles will be the right angle.
8. Each diagonal bisects the parallelogram into two congruent triangles.
9. The sum of the square of all the sides of a parallelogram is equal to the sum of the square of its diagonals. It is also called parallelogram law.

Converse Properties of Parallelogram

The converse of the parallelogram properties are given below. A Quadrilateral is a Parallelogram if:

1. One pair of opposite sides are equal and parallel.
2. Opposite sides are equal
3. Opposite angles are equal
4. Diagonals bisect each other

Perimeter and Area of a Parallelogram:

The perimeter of a Parallelogram: The sum of all the sides of a parallelogram is the perimeter of a Parallelogram.

Perimeter of a Parallelogram \(P = 2\left({{\text{Length}}+{\text{Breadth}}} \right)\,{\text{units}}\)

Area of a Parallelogram: The region occupied by the Parallelogram is the area of a Parallelogram.

Area of a Parallelogram, \(A = \left({{\text{base}} \times {\text{height}}} \right)\,{\text{square}}\,{\text{units}}\)

Special Types of Parallelograms

Special types of Parallelograms are explained below.

Square: A parallelogram having all sides equal and each angle measure equal to one right angle is called a square.

Properties of a square:

1. All four sides are equal in a square.
2. Each angle in a square is a right angle.
3. The diagonals are of equal length.
4. The diagonals bisect each other at right angles.

Rhombus: A parallelogram having all sides equal is called a rhombus and opposite angles are equal.

Properties of Rhombus:

1. All the sides of a rhombus are equal.
2. The opposite angles are equal.
3. The adjacent angles of a rhombus are adding up to \(180\) degrees.
4. The diagonals of a rhombus bisect each other at \(90°\).

Rectangle: A parallelogram whose each angle is \(90°\) is called a rectangle.

Properties of a Rectangle:

1. Each angle of a rectangle is \(90°\).
2. The lengths of the diagonals in a rectangle are equal.
3. The length of the adjacent sides are unequal, but the lengths of the opposite sides are equal.

Note: Since a rhombus, a rectangle and a square are special types of a parallelogram, so all the properties of a parallelogram are true for each of them.

Properties of a Trapezium

1. Trapezium has four sides. The sides of the trapezium \(ABCD\) are \(AB, BC, CD, DA\).

2. Trapezium has one pair of parallel and another pair of non-parallel sides. Parallel sides are \(AB\) and \(DC\), non-parallel sides are \(AD\) and \(BC\).

3. Trapezium has \(4\) angles. 

Four angles of a trapezium \(ABCD\) are Angles \(∠A, ∠B, ∠C, ∠D\).

4. The sum of all the four angles is equal to \(360°\)

 \(∠A+∠B+∠C+∠D = 360°\).

5. Two pairs of adjacent angles of a trapezium formed between the parallel side and non-parallel side, sum up to\(180°\).

\(∠A+∠C = 180°=∠B+∠D\).

6. In a regular trapezium the diagonals bisect each other.

7. Exactly one pair of sides are parallel.

8. The legs of the isosceles trapezium are congruent.

9. Non-parallel sides are unequal, except in isosceles trapezium.

10. Parallel sides of the trapezium are bases, and the non-parallel sides are known legs.

Solved Examples – Properties of a Parallelogram

Q.1.Find the area of a parallelogram whose base is \(12\,{\text{cm}}\) and height is \(5\,{\text{cm}}\).
Ans:
Given:
Base \(b\) of a parallelogram \(= 12\,{\text{cm}}\)
Height \(h\) of a parallelogram \(= 5\,{\text{cm}}\)
We know that area of a parallelogram \(= {\text{base}} \times {\text{height}}\)
Area \(A\) of a parallelogram \(12 \times 5 = 60\,{\text{c}}{{\text{m}}^2}\)
Hence, the area of the given parallelogram is \(60\,{\text{c}}{{\text{m}}^2}\).

Q.2.The perimeter of a parallelogram-shaped plot is \(400\,{\text{m}}\) and one of its adjacent sides is \(150\,{\text{m}}\). Find the measure of its another adjacent side.
Ans:
Given:
The perimeter of a parallelogram shaped plot \( = 400\,{\rm{m}}\)
One of the adjacent sides of a parallelogram shaped plot \( = 150\,{\rm{m}}\)
Let another adjacent side of a parallelogram shaped plot be \(x\) meter
We know that perimeter of a parallelogram \(= 2\left( {{\rm{sum}}\,{\rm{of}}\,{\rm{adjacent}}\,{\rm{sides}}} \right)\)
So, \(400\,m=2×(150+x)\,m\)
\(\Rightarrow 200\,{\text{m}} = 150 + {\text{x}}\,{\text{m}}\)
\(\Rightarrow x = 50\,{\text{m}}\)
Therefore, another adjacent side of a given parallelogram is \(50\,{\rm{m}}.\)

Q.3. The area of a parallelogram-shaped plot is \(7500\,{{\text{m}}^2}\) and its altitude is \(75\,{\text{m}}\). Find the base of a parallelogram-shaped plot.
Ans:
Given:
Area of a Parallelogram shaped plot \(= 7500\,{{\rm{m}}^2}\)
Altitude (h) of a Parallelogram shaped plot \(= {\rm{ }}75,\,{\rm{m}}\)
We know that area of a parallelogram \( = {\text{Base}} \times {\text{Height}}\)
\(\Rightarrow 7500\,{{\rm{m}}^2} = b \times 75\,{\rm{m}}\)
\(\Rightarrow b = \frac{{7500}}{{75}} = 100\,{\rm{m}}\)
Hence, the base of a given parallelogram shaped plot is \(100\,{\text{m}}\).

Q.4. Area of a parallelogram shaped park is \(6250\,{{\text{m}}^2}\) and its base is \(250\,{\text{m}}\). Find the altitude of a parallelogram shaped park.
Ans:
Given:
Area of a parallelogram-shaped park \(= 6250\,{{\rm{m}}^2}\)  
Base b of a parallelogram-shaped park \(= 250\,{\rm{m}}\)
We know that area of a parallelogram \(= {\rm{Base}}\, \times \,{\rm{Height}}\)
\(⟹6250 =250 ×h\)
\(\Rightarrow h = \frac{{6250}}{{250}} = 25\,{\rm{m}}\)
Hence, the altitude of a given parallelogram shaped park is \(25\,{\text{m}}\).

Q.5. The adjacent sides of a parallelogram are \(15\,{\text{cm}}\) and \(8\,{\text{cm}}\). Find the perimeter of a parallelogram?
Ans:
Let the adjacent sides of the given parallelogram be \(a\) & \(b\).
So, \(a = 15\,{\text{cm}}\) and \(b = 8\,{\text{cm}}\)
We know that perimeter of a parallelogram \(=2\left({{\text{sum}}\,{\text{of}}\,{\text{adjacent}}\,{\text{sides}}}\right)\)
Now, the perimeter of a given parallelogram \(=2(a+b)=2(15+8)\)
\( \Rightarrow 2 \times 23 = 46\,{\text{cm}}\)
Hence, the perimeter of a given Parallelogram is \(46\,{\text{cm}}\).

Summary

A Parallelogram is one of the geometrical shapes with two -dimensions. A parallelogram is a quadrilateral with four sides in which the opposite sides are parallel to each other. The length of the pair of opposite sides is equal. Building structures, rangoli designs, tiles, tables, erasers etc., are the few things in which we can commonly see Parallelogram shape.

In this article, we have learned that a parallelogram is a polygon whose opposite sides are equal and parallel. The properties which are applicable to the sides, angles, diagonals, vertices of a parallelogram are the properties of a parallelogram. Also, the article gave insights about the definition of a parallelogram, properties of a parallelogram, formula to find the area and perimeter of a parallelogram. The learning outcome from the properties of a parallelogram article will help in understanding all the properties of parallelogram clearly and it will help apply the properties in solving many mathematical problems related to geometry. This knowledge will also be useful in real life.

Frequently Asked Questions (FAQ) – Properties of a Parallelogram

Q.1. What are diagonals in a parallelogram?
Ans: The line segments which joins the opposite vertices of the parallelogram are diagonals in a parallelogram. A parallelogram has two diagonals.

Q.2. What are the \(4\) properties of a parallelogram?
Ans: \(4\) properties of a parallelogram are

1. The opposite sides are congruent and parallel.
2. The opposite angles are equal.
3. The adjacent angles are supplementary.
4. If anyone of the angles is \(90°\), then all the other angles will be \(90°\).

Q.3. What are the \(7\) properties of a parallelogram?
Ans: \(7\) properties of a parallelogram are

1. A quadrilateral is called a parallelogram, if both pairs of its opposite sides are parallel and equal.
2. The diagonals are bisecting each other.
3. The opposite angles are of equal measure.
4. The pair of adjacent angles are supplementary.
5. The opposite sides are parallel and congruent.
6. The opposite angles are congruent.
7. The consecutive angles are supplementary.

Q.4. Does a parallelogram have equal diagonals?
Ans: Yes, a parallelogram can have equal diagonals only in the case of special parallelograms in square and in a rectangle. But the diagonals are not equal in other types of parallelograms.

Q.5. What are \(3\) of \(8\) properties of a parallelogram?
Ans: \(3\) properties of a parallelogram are

1. In a parallelogram, the diagonals are bisecting each other.
2. The opposite angles are equal.
3. The pair of adjacent angles add up to \(180\) degrees.

We hope you find this article on Properties of a Parallelogram helpful. In case of any queries, you can reach back to us in the comments section, and we will try to solve them.