# Parallelograms

 1 Introduction to Parallelograms 2 What is a Parallelogram? 3 Types of Parallelograms 4 Properties of a Parallelogram 5 Parallelogram Theorems 6 Area of a Parallelogram 7 Solved Examples on Parallelograms 8 Thinking out of the Box! 9 Practice Questions on Parallelograms 10 Challenging Questions on Parallelograms 11 Maths Olympiad Sample Papers 12 Frequently Asked Questions (FAQs) 13 FREE Worksheets on Parallelograms

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## Introduction to Parallelogram

A parallelogram is a special kind of quadrilateral

Rectangle, square, and rhombus are parallelogram examples. ### Parallelogram: Definition

A parallelogram is defined as a quadrilateral in which both pairs of opposite sides are parallel and equal.

## What is a Parallelogram?

A parallelogram is a convex polygon with $$4$$ edges and $$4$$ vertices.

It is a type of quadrilateral where the opposite sides are parallel and equal.

Drag the vertices to understand the relations between different elements of a parallelogram.

## Types of Parallelograms

Rhombuses, rectangles, and squares are common parallelogram examples.

The parallelogram properties for each are listed below.

### Rectangle A rectangle has:

• two pairs of parallel sides$A B\|D C \text{ and } A D\| B C$
• four right angles$\angle A=\angle B=\angle C=\angle D=90^{\circ}$
• opposite sides of equal lengths$A B=D C \text{ and } A D= B C$
• two equal diagonals$AC=BD$
• diagonals that bisect each other i.e., one diagonal divides the other diagonal into exactly two halves.

### Square A square has:

• four equal sides$AB=BC=CD=DA$
• four right angles$\angle A=\angle B=\angle C=\angle D=90^{\circ}$
• two pairs of parallel sides$A B\|D C \text { and } A D\| B C$
• two equal diagonals$AC=BD$
• diagonals that are perpendicular to each other$AC \perp BD$
• diagonals that bisect each other. i.e., one diagonal divides the other diagonal into exactly two halves.

### Rhombus A rhombus has:

• two pairs of parallel sides$E H\|F G \text{ and } E F\| H G$
• four equal sides$EH=HG=GF=FE$
• opposite angles are equal$\angle E = \angle G \text{ and } \angle H = \angle F$
• diagonals that are perpendicular to each other$EG \perp HF$
• diagonals that bisect each other. i.e., one diagonal divides the other diagonal into exactly two halves.

## Properties of a Parallelogram

Consider the parallelogram $$\text{PQRT}$$.

Let us understand the various parallelogram properties using $$\text{PQRT}$$. • The opposites sides of a parallelogram are parallel.
$P Q\|R T \text{ and } PR\| Q T$
• Opposite sides of a parallelogram are equal.
$P Q=R T \text{ and } PR= Q T$
• Opposite angles of a parallelogram are equal.
$\angle P = \angle T \text{ and }\angle Q = \angle R$
• Diagonals of a parallelogram bisect each other. i.e., one diagonal divides the other diagonal into exactly two halves.
$R E=E Q \text{ and } PE= ET$
• Same-side interior angles supplements each other.
$\angle PRT + \angle RTQ =180 ^\circ$
$\angle RTQ + \angle TQP =180 ^\circ$
$\angle TQP + \angle QPR =180 ^\circ$
$\angle QPR + \angle PRT =180 ^\circ$
• Diagonals divide the parallelogram into two congruent triangles.
$\Delta \text{RPQ} \: \text{congruent to} \:\Delta \text{QTR}$
$\Delta \text{RPT} \: \text{congruent to}\: \Delta \text{QTP}$

## Parallelogram Theorems

### Theorem 1

In a parallelogram, opposite sides are equal. Conversely, if the opposite sides in a quadrilateral are equal, then it is a parallelogram.

Consider the following figure: Proof:

In $$\Delta ABC$$ and $$\Delta CDA$$,

\begin{align} AC&=AC \;(\text{common sides}) \\ \angle 1&=\angle 4 \;(\text{alternate interior angles}) \\ \angle 2&=\angle 3 \;(\text{alternate interior angles}) \end{align}

Thus, by the ASA criterion, the two triangles are congruent, which means that the corresponding sides must be equal.

Thus,

\begin{align}\boxed{AB=DC\;\text{and}\;AD=BC} \end{align}

This proves that opposite sides are equal in a parallelogram.

### Theorem 2

In a parallelogram, opposite angles are equal. Conversely, if the opposite angles in a quadrilateral are equal, then it is a parallelogram.

Consider the following figure: Proof:

In $$\Delta ABC$$ and $$\Delta CDA$$,

\begin{align} AC&=AC\: ( \text{common sides} )\\ \angle 1&=\angle 4 \:(\text{alternate interior angles}) \\ \angle 2&=\angle 3 \: (\text{alternate interior angles} ) \end{align}

Thus, the two triangles are congruent, which means that

\begin{align}\boxed{\angle B=\angle D} \end{align}

Similarly, we can show that

\begin{align}\boxed{\angle A=\angle C} \end{align}

This proves that opposite angles in any parallelogram are equal.

### Theorem 3

In a parallelogram, the diagonals bisect each other. Conversely, if the diagonals in a quadrilateral bisect each other, then it is a parallelogram.

Consider the following figure: Proof:

In $$\Delta AEB$$ and $$\Delta DEC$$, we have:

\begin{align} AB&=CD \:( \text{opposite sides of a parallelogram})\\ \angle 1&=\angle 3\:( \text{alternate interior angles}) \\ \angle 2&=\angle 4\:( \text{alternate interior angles}) \end{align}

By the ASA criterion, the two triangles are congruent, which means that

\begin{align}\boxed{AE=EC\;\text{and}\;BE=ED}\end{align}

Thus, the two diagonals bisect each other.

### Theorem 4

In a quadrilateral, if one pair of opposite sides are equal and parallel, then it is a parallelogram.

Consider the following figure: It is given that $$AB=CD$$ and $$AB || CD$$.

We have to prove that ABCD is a parallelogram.

Proof:

In $$\Delta AEB$$ and $$\Delta DEC$$, we have:

\begin{align} \text{AB}&=\text{CD} \: \text{given} \\ \angle 1&=\angle 3\: \text{alternate interior angles} \\ \angle 2&=\angle 4\: \text{alternate interior angles} \end{align}

Thus, the two triangles are congruent, which means that:

\begin{align}\boxed{AE=EC\;\text{and}\;BE=ED}\end{align}

Therefore, the diagonals AC and BD bisect each other, and this further means that ABCD is a parallelogram.

## Area of a Parallelogram Consider the parallelogram having base (b) and height (h).

Parallelogram area is given by the formula:

 $$\text{Area of Parallelogram} \!=\!\text{Base} \!\times\! \text{Height}$$

Input the base and height in the parallelogram area calculator shown below to calculate the area of a parallelogram.

Note: Parallelogram perimeter formula:

 $$2 \times$$ (Sum of length of adjacent sides)

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## Solved Examples

 Example 1

In the given parallelogram$$\text{ABCD}$$, $$\angle \text{A} = 90^\circ$$.

Show that all its angles are equal to $$90^\circ$$. Solution:

Applying the parallelogram properties we have learnt, we know that:

• the opposite angles are equal

\begin{align} \therefore \angle \text{A} = \angle \text{C} = 90^\circ \end{align}

• the adjacent angles are supplementary

\begin{align} \angle \text{A} +\angle \text{D} &= 180^\circ \\ \angle\text{D} &= 180^\circ -90^\circ (\because \angle\text{A} = 90^\circ \: \text{given} )\\ \therefore \angle\text{D} &= 90^\circ\ \\ \Rightarrow \angle \text{B} &=90^\circ \end{align}

 \begin{align}\angle\text{A}=\angle\text{B}=\angle\text{C}=\angle\text{D}= 90^\circ\end{align}
 Example 2

Prove that when the parallelogram diagonals bisect each other at $$90^\circ$$, it is a rhombus.

Solution:

Consider the parallelogram $$\text{ABCD}$$. \begin{align} \Delta \text{AEB}\:\text{and}\: \Delta \text{AED} \\ AE&=AE\: (\text{common})\\ BE&=ED \:(\text{given})\\ \angle AEB&=\angle AED=\,90^\circ\ (\text{given} ) \end{align}

Therefore, by SAS Congruency, $$\Delta AEB$$ and $$\Delta AED$$ are congruent.

\begin{align} \Rightarrow \text{AB = AD} \end{align}

Similarly,

Considering  $$\Delta \text{AED}\:\text{and}\: \Delta \text{CED}$$

\begin{align} \Rightarrow \text{AD = DC} \end{align}

Which further implies,

\begin{align}\boxed{ AB=BC=CD=AD} \end{align}

 $$\therefore$$The given parallelogram is a rhombus.
 Example 3

If in a parallelogram $$\text{ABCD}$$, $$\text{AC}$$ bisects  $$\angle\text{A}$$ and $$\angle\text{C}$$, show that $$\text{AC} \perp \text{BD}$$.

Solution: \begin{align} \Delta \text{ABC}\:and\: \Delta \text{ADC}: \\ AC&=AC (\text{common}) \\ \angle 1&=\angle 2 \:( \text{given)} \\ \angle 3 &=\angle 4 \:( \text{given}) \end{align}

By the ASA criterion, the two triangles are congruent. This means that

\begin{align} \Rightarrow \text{AB = AD} \end{align}

\begin{align} \Delta \text{AEB}\:and\: \Delta \text{AED}: \\ AE&=AE (\text{common})\\ AB&=AD (\text{proved above}) \\ \angle 2&=\angle 1 \:( \text{given)} \\ \end{align}

By the SAS criterion, the two triangles are congruent.

Therefore,

\begin{align} \angle AEB = &\angle AED\\ = &\frac{1}{2} \times 180^\circ \\= &\,90^\circ \end{align}

Hence,

\begin{align}\boxed{\angle AEB=\angle AED = 90^\circ} \end{align}

 $$\therefore \;\text{AC} \perp\ \text{BD}$$
 Example 4

Find the area of a parallelogram whose base is $$7\:\text{cm}$$ and height is $$10\:\text{cm}$$.

Solution:

Given Base $$=7\:\text{cm}$$

Height$$=10\:\text{cm}$$

Area of a parallelogram

\begin{align} &=\text{base} \times \text{height} \\ &= 7 \times 10 \\ &=70\: \text{cm}^2 \end{align}

 $$\therefore$$ Area of the parallelogram $$=70\: \text{cm}^2$$ Think Tank
1. Can a kite be called a parallelogram?
2. What elements of a trapezium should be changed to make it a parallelogram?
3. Can there be a concave rhombus?

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## Practice Questions

Here are few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result. Challenging Questions
1. Two parallel lines are intersected by a transversal. Consider a closed four-sided figure ABCD formed by the bisectors of the interior angles. Prove that ABCD is a rectangle. IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

## 1. What are the different types of parallelograms?

Rectangle, rhombus and square are the three different types of parallelograms.

## 2. What are the various parallelogram formulas?

The various formulas for parallelograms are given here:

1. Area of the parallelogram when the base and height is known:

$$\text{base} \times \text{height}$$

2. Area of the parallelogram when the diagonals are known:

$$\frac{1}{2} \times d_{1} \times d_{2} sin (y)$$

where $$y$$ is the angle at the intersection of the diagonals

3. Area of the parallelogram using Trignometry:

$$\text{ab}$$$$sin(x)$$

where $$\text{a}$$ and $$\text{b}$$ are the length of the parallel sides and $$x$$ is the angle between the given sides of the parallelogram.

Parallelogram perimeter:

$$2 \!\times\! \text{(sum of length of adjacent sides)}$$

## 3. What is the area of a parallelogram?

The area of a parallelogram is the space occupied by the parallelogram in two-dimensional space.

grade 9 | Questions Set 1