Quadrilaterals

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 shape with four vertices and four sides.

Two examples are shown below.

We can observe that a quadrilateral has four sides, four angles, and four vertices.

When we have to name a quadrilateral, we have to keep in mind the order of the vertices.

For example, consider the following quadrilateral:

We could name this quadrilateral \(ABCD, BCDA, ADCB, DCBA\), etc, but we cannot name it \(ACBD, BACD\), etc. since in the latter names, the order of the vertices is wrong.

A quadrilateral ABCD will have two diagonals: \(AC\) and \(BD\).

There are different types of quadrilaterals.

Here are a few types of quadrilateral shapes:

1. Square
2. Rectangle
3. Parallelogram
4. Trapezium
5. Rhombus
6. Kite

The common properties of quadrilaterals are:

• They have four sides.
• They have four vertices.
• They have two diagonals.
• The sum of all interior angles is \(360^\circ\).

We will study about the other properties of different quadrilateral shapes in detail.

We can identify a quadrilateral by using the following properties of quadrilaterals.

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 exactly two halves.

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 exactly two halves.

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.

Trapezium

A trapezium 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.

There is nothing special about the sides, angles, or diagonals of a trapezium.

But if the two non-parallel opposite sides are of equal length, then it is called an isosceles trapezium.

The above quadrilateral \(XYZW\) is an isosceles trapezium.

In an isosceles trapezium, the lengths of the diagonals are equal. i.e., \(XZ=WY\).

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 exactly two halves.

Kite

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

A kite has:

• two pairs of equal adjacent sides
  \[AB=BC \text{ and }CD=DA\]
• one pair of opposite angles (which are obtuse) that are equal
  \[\angle A = \angle C\]
• diagonals that are perpendicular to each other
  \[AC \perp BD\]
• a longer diagonal that bisects the shorter diagonal.

We can understand the hierarchy of quadrilaterals using the following chart.

From the above figure, we can note certain key points. For example:

• All squares are rectangles, but not all rectangles are squares.
• All squares are rhombuses, but not all rhombuses are squares.

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1. Any four-sided figure is a quadrilateral.

For example, the figure given below is a quadrilateral.

Though the above figure has 4 sides and is a closed figure, it is NOT a quadrilateral.

Quadrilaterals are closed figures made of non-intersecting line segments.  

2. All rectangles are squares.

Many rectangles do not have 4 equal sides and 4 equal angles. Therefore, all rectangles are NOT squares.

However, all squares are rectangles. As they all have 4 right angles, their opposite sides are parallel to each other and the opposite sides are equal to each other.

3. All rhombuses are squares.

Many rhombuses do not have four equal sides and four equal angles. Thus, all rhombuses are NOT squares.

However, all squares are rhombuses. As they have 4 equal sides, their opposite sides are parallel to each other.

The area of a quadrilateral is the number of unit squares that can be fit into it.

The following table lists the formulas for finding the area of quadrilaterals.

Quadrilateral	Figure	Area
Square		\(x \times x\)
Rectangle		\(l \times b\)
Parallelogram		\(b \times h\)
Isosceles Trapezium		\( \frac{1}{2}(a+b)h \)
Rhombus		\( \frac{1}{2} \times d_1 \times d_2 \)
Kite		\( \frac{1}{2} \times d_1 \times d_2 \)

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Example 1

 

 

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 2

 

 

What is the area of a kite whose diagonals are of length \(10\) cm and \(20\) cm?

Solution:

The diagonals of the given kite are:

\[d_1=10\\d_2=20\]

The area of the kite is calculated using the formula:

\[\begin{align}
A &= \dfrac{1}{2} \times d_1 \times d_2\\[0.3cm]
&= \dfrac{1}{2}\times 10 \times 20\\[0.3cm]
&= 100
\end{align} \]

\(\therefore  \text{Area of the kite}=100  \text{ cm}^2\)

Example 3

 

 

What is the area of the following isosceles trapezium?

Solution:

The bases of the given isosceles trapezium are:

\[a=16\\b=40\]

The area of an isosceles trapezium is calculated using the formula:

\[\begin{align}
 \text{Area of an isosceles trapezium }&= \frac{1}{2}(a+b)h \\[0.3cm]
&=\frac{1}{2} (16+40)15\\[0.3cm]
&=\frac{1}{2} (840)\\[0.3cm]
&= 420\end{align} \]

\(\therefore  \text{Area of the trapezium}=420\text{ unit}^2\)