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What is a Parallelogram – Definition, Properties & Examples

properties of parallelogram

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The word ‘parallelogram’ is derived from the Greek word ‘parallelogrammon’ which means “bounded by parallel lines”. A parallelogram is a quadrilateral that is bounded by parallel lines. It is a 2D shape in which the opposite sides are parallel and equal. 

There are three types of parallelograms – square, rectangle, and rhombus, and each of them has its own unique properties.

Let’s understand what is a parallelogram and the properties of parallelogram.

What is a Parallelogram?

A parallelogram is a special type of quadrilateral that is formed by parallel lines. In a parallelogram, both pairs of opposite sides are parallel and equal. Hence, a parallelogram is defined as a quadrilateral in which both pairs of opposite sides are parallel and equal. 

The above figure shows three types of parallelograms.

  • Rectangle 
  • Square
  • Rhombus
properties of parallelogram

Properties of a Parallelogram

The following are the basic properties of parallelograms that help you identify them. 

  • The opposite sides of a parallelogram are parallel. Here, $\text{AB} ‖ \text{CD}$ and $\text{BC} ‖ \text{DA}$.
  • The opposite sides of a parallelogram are equal. In the above figure, $\text{AB} = \text{CD}$ and $\text{BC} = \text{DA}$.
  • The opposite angles of a parallelogram are equal. Here, $\angle \text{A} = \angle \text{C}$ and $\angle \text{B} = \angle \text{D}$
  • The diagonals of a parallelogram bisect each other. Here, $\text{AC} = \text{BD}$
  • Adjacent angles are supplementary. In the above figure, $\angle \text{A} + \angle \text{B} = 180^{\circ}$, $\angle \text{B} + \angle \text{C} = 180^{\circ}$, $\angle \text{C} + \angle \text{D} = 180^{\circ}$ and $\angle \text{D} + \angle \text{A} = 180^{\circ}$. 
  • The diagonals divide the parallelogram into two congruent triangles. Here, $\triangle \text{ABD} \cong \triangle \text{BCD}$, and $\triangle \text{ABC} \cong \triangle \text{ACD}$.
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Diagonals of a Parallelogram Are Equal

The diagonals of a parallelogram divide it into two congruent triangles, i.e., in a parallelogram $\text{ABCD}$, $\triangle \text{ABD} \cong \triangle \text{BCD}$, and $\triangle \text{ABC} \cong \triangle \text{ACD}$.

properties of parallelogram

Let’s see how to prove the above statement.

Since $\text{ABCD}$ is a parallelogram, the opposite sides are equal.

Therefore, $\text{AB} = \text{CD}$ and $\text{BC} = \text{DA}$.

Now, in $\triangle \text{ABD}$ and $\triangle \text{CBD}$

$\text{AB} = \text{CD}$ (Opposite sides of a parallelogram)

$\text{DA} = \text{BC}$ (Opposite sides of a parallelogram)

$\text{BD} = \text{BD}$ ( common)

Thus, $\triangle \text{ABD} \cong \triangle \text{CBD}$ (SSS congruency criterion).

Therefore, we can say that the diagonal of a parallelogram divides it into two congruent triangles.

Opposite Sides of a Parallelogram Are Equal

The opposite sides of a parallelogram are equal, i.e., in a parallelogram $text{ABCD}$, $\text{AB} = \text{CD}$, and $\text{BC} = \text{DA}$.

properties of parallelogram

Let’s see how to prove the above statement.

In the above figure, $\text{ABCD}$ is a parallelogram, and $\text{AC}$ is one of the diagonals. 

The diagonal $\text{AC}$ divides parallelogram $\text{ABCD}$ into two triangles, namely, $\triangle \text{ABC}$ and $\triangle \text{ABC}$. 

In order to prove that opposite sides are equal i.e., $\text{AB} = \text{CD}$, and $\text{BC} = \text{DA}$, we need to first prove that $\triangle \text{ABC}  \cong \triangle \text{ABC}$ 

In $\triangle \text{ABC}$ and $\triangle \text{CDA}$, $\text{BC} || \text{DA}$ and $\text{AC}$ is a transversal.

So, $\angle \text{BCA} = \angle \text{DAC}$ (Pair of alternate angles)

And $\text{AC} = \text{CA}$ (common)

Thus, $\triangle \text{ABC} \cong \triangle \text{CDA}$ (ASA rule).

Therefore, the corresponding parts $\text{AB} = \text{CD}$ and $\text{DA}= \text{BC}$.

Opposite Angles of a Parallelogram Are Equal

In a parallelogram the opposite angles are equal. 

Let’s see how to prove the above statement, i.e, in a parallelogram $\text{ABCD}$, $\angle \text{A} = \angle \text{C}$, and $\angle \text{B} = \angle \text{D}$.

properties of parallelogram

In the above figure, in a parallelogram $\text{ABCD}$, $\text{AB} || \text{CD}$ and $\text{AD} || \text{BC}$.

Consider $\triangle \text{ABC}$ and $\triangle \text{ADC}$

$\text{AC} = \text{AC}$ (common side)

We know that alternate interior angles are equal.

$\angle \text{BAC} = \angle \text{ACD}$

$\angle \text{BCA} = \angle \text{CAD}$

Therefore, $\triangle \text{ABC} \cong \triangle \text{ADC}$

Hence, $\angle \text{A} = \angle \text{C}$, and $\angle \text{B} = \angle \text{D}$. (Corresponding Parts of Congruent Triangles).

Adjacent Angles of a Parallelogram Are Supplementary

In a parallelogram the adjacent angles are supplementary.

properties of parallelogram

Let’s see how to prove the above statement, i.e, in a parallelogram $\text{ABCD}$, $\angle \text{A} + \angle \text{B} = 180^{\circ}$, and $\angle \text{C} + \angle \text{D} = 180^{\circ}$.

In the above figure, $\text{AB} ∥ \text{CD}$ and $\text{AD}$ is a transversal.

We know that interior angles on the same side of a transversal are supplementary.

Therefore, $\angle \text{A} + \angle \text{D} = 180^{\circ}$

Similarly, $\angle \text{B} + \angle \text{C} = 180^{\circ}$, $\angle \text{C} + \angle \text{D} = 180^{\circ}$ and $\angle \text{A} + \angle \text{B} = 180^{\circ}$.

Therefore, the sum of any two adjacent angles of a parallelogram is equal to $180^{\circ}$.

Greatest Math Discoveries

Practice Problems

  1. Define parallelogram.
  2. State True or False
    • Adjacent sides of a parallelogram are equal
    • Adjacent sides of a parallelogram are parallel
    • Opposite sides of a parallelogram are equal
    • Opposite sides of a parallelogram are parallel
    • Adjacent angles of a parallelogram are equal
    • Adjacent angles of a parallelogram are supplementary
    • Opposite angles of a parallelogram are equal
    • Opposite angles of a parallelogram are supplementary

FAQs

What is a parallelogram in geometry?

In geometry, a parallelogram is a quadrilateral (4-sided 2D shape) that has its opposite sides parallel and equal in length.

Are all the angles of a parallelogram equal?

No, all the angles of a parallelogram are not equal. Only the opposite angles of a parallelogram are equal, whereas the adjacent angles of a parallelogram are supplementary, i.e., their sum is $180^{\circ}$.

What is the difference between a parallelogram and a quadrilateral?

All parallelograms are quadrilaterals but all quadrilaterals are not necessarily parallelograms. For example, a trapezium is a quadrilateral, but not a parallelogram. For a quadrilateral to be a parallelogram, all the opposite sides must be parallel and equal to each other.

Is a rhombus a parallelogram?

Yes, a rhombus is a parallelogram in which the opposite sides are parallel and the opposite angles are congruent. Apart from this, all the sides of a rhombus are equal and the diagonals bisect each other at right angles.

Is a trapezium a parallelogram?

No, a trapezium is not a parallelogram, since all opposite sides of the trapezium are not parallel to each other. A trapezium has only one pair of opposite sides parallel to each other. Also, a trapezium doesn’t have opposite sides equal to each other. Hence, it is a quadrilateral but not a parallelogram.

Conclusion

A parallelogram is a special type of quadrilateral that is formed by parallel lines. In a parallelogram, both pairs of opposite sides are parallel and equal. A parallelogram has certain unique properties that distinguish it from the other quadrilaterals. The three types of parallelograms are square, rectangle, and rhombus.

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