# Some Particular Types of Quadrilaterals

We have heard and read about squares and rectangles, which are known as polygons with 4 sides and 4 vertices.

They are also known as quadrilaterals.

However, do you know that there are other types of quadrilaterals too?

Once upon a time, there were 4 children who were flying kites in a garden.

They looked at the shapes of their kites very carefully and found out that all the four kites have 4 sides and 4 vertices.

Let's learn about some particular types of quadrilaterals in this lesson by learning about their names and properties with the help of some solved examples and a few interactive questions for you to test your understanding.

Let's begin!

## Lesson Plan

 1 What Are the Particular Types of Quadrilaterals? 2 Important Notes 3 Thinking Out of the Box! 4 Solved Examples 5 Interactive Questions

The word "quadrilateral" is derived from two Latin words, "Quadri" which means "four," and "Latus" which means "side."

Hence, a quadrilateral is a polygon with 4 sides.

A quadrilateral can be defined in two ways:

• A quadrilateral is a closed shape that is obtained by joining four points among which any three points are non-collinear.
• A quadrilateral is a closed 2D shape with four vertices and four sides.

Observe the two examples shown below.

## What Are the Particular Types of Quadrilaterals?

There are majorly six types of quadrilaterals.

Observe the relationship between each type of quadrilateral from the given chart.

Observe the table given below to have a basic idea of the shapes of different types of quadrilaterals.

 S.No. Quadrilateral Shape 1 Square 2 Rectangle 3 Parallelogram 4 Trapezoid 5 Rhombus 6 Kite

Important Notes
1. Every square is a rectangle, but every rectangle is not a square.
2. Every square is a rhombus, but every rhombus is not a square.
3. Squares, rectangles, and rhombuses come under the category of parallelograms.

## Properties of Particular Types of Quadrilaterals

The properties of quadrilaterals are listed below:

• They have four sides.
• They have a 2D shape.
• They have four vertices.
• They have two diagonals.
• The sum of all the interior angles of a quadrilateral is $$360^\circ$$.
Now, let us see the properties of particular types of quadrilaterals

### Properties of a Square

A square is a quadrilateral with four equal sides and four right angles.

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 two equal halves.

### Properties of a Rectangle

A rectangle is a quadrilateral with two pairs of equal and parallel opposite sides and four right angles.

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 two equal halves.

### Properties of a Parallelogram

A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.

A parallelogram has:

• Two pairs of parallel sides
$P Q\|R T \text{ and } PR\| Q T$
• Opposite sides of equal lengths
$P Q=R T \text{ and } PR= Q T$
• Opposite angles that are equal
$\angle P = \angle T \text{ and }\angle Q = \angle R$
• Two diagonals that bisect each other, i.e., one diagonal divides the other diagonal into exactly two halves.

### Properties of a Trapezoid

A trapezoid is a quadrilateral in which one pair of opposite sides is parallel.

• The sides that are parallel to each other are called bases.
In the above figure, $$EF$$ and $$GH$$ are bases.
• The sides that are not parallel to each other are called legs.
In the above figure, $$EG$$ and $$FH$$ are legs.

If the two non-parallel opposite sides of a trapezoid are of equal length, then it is called an isosceles trapezoid.

The above quadrilateral $$XYZW$$ is an isosceles trapezoid.

In an isosceles trapezoid, the lengths of the diagonals are equal. i.e., $$XZ=WY$$ and the base angles are congruent.

### Properties of a Rhombus

A rhombus is a quadrilateral with four equal sides.

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 two equal halves.

### Properties of a Kite

A kite is a quadrilateral in which two pairs of adjacent sides are equal.

A kite has:

• Two pairs of equal adjacent sides
$AC=BC \text{ and }AD=BD$
• One pair of opposite angles (which are obtuse) that are equal
$\angle A = \angle B$
• Diagonals that are perpendicular to each other
$AB \perp CD$
• A longer diagonal that bisects the shorter diagonal.

Think Tank
1. Can a kite be called a parallelogram?
2. What elements of a trapezoid should be changed to make it a parallelogram?

## Solved Examples

 Example 1

Cindy knows that the diagonals of a parallelogram bisect each other. If they bisect each other

at $$90^\circ$$, does it become 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\ ) \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}

This further implies,

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

We know that the sides of a rhombus are equal in length.

 $$\therefore$$ The given parallelogram is a rhombus.
 Example 2

Can you find the angle $$x^\circ$$ in the following figure?

Solution

We know that the sum of the angles in a quadrilateral is $$360^\circ$$.

From the given figure, we get:

\begin{align} x+67+77+101 &=360\\[0.3cm] x+245&=360\\[0.3cm] x &=115 \end{align}

 $$\therefore x^\circ = 115^\circ$$
 Example 3

Identify the pairs of equal sides in the kite given below.

Solution

We know that a kite has two pairs of equal adjacent sides.

The pairs of adjacent sides in the above kite are (PQ, QR), (PQ, PS), (QR, RS), and (PS, RS)

Pairs of equal adjacent sides are (PQ, QR) and (PS, RS)

 $$\therefore$$ Pairs of equal sides are (PQ, QR) and (PS, RS).

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

The mini-lesson targeted the fascinating concept of some particular types of quadrilaterals. The math journey around some particular types of quadrilaterals starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

### 1. Is rhombus a kite?

Yes, a rhombus is a kite because the adjacent sides in a rhombus are equal.

Quadrilaterals are 2D closed shapes having 4 sides and 4 vertices.

### 3. What are the 8 types of quadrilaterals?

The eight types of quadrilaterals are square, rectangle, parallelogram, rhombus, trapezoid, isosceles trapezoid, kite, and irregular quadrilaterals.

### 4. Can we say that diamond shape is a quadrilateral?

Yes, we usually represent the shape of a diamond with a rhombus or a kite. So, a diamond shape is a quadrilateral.

### 5. Is every square a rectangle?

Yes, every square is a rectangle as it satisfies all the properties of a rectangle.

### 6. Is every rhombus a parallelogram?

Yes, every rhombus is a parallelogram because its opposite sides are parallel and equal; the opposite angles are equal and the diagonals bisect each other.

### 7. Can a parallelogram be considered as a rectangle?

A parallelogram can be considered as a rectangle only when all four interior angles in a parallelogram are of 90° each.

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