# Quadrilaterals and their properties

## What is a quadrilateral?

A quadrilateral is a polygon with 4 sides and 4 vertices.

### Types of quadrilaterals

**Square**

Properties of a square

A square has

- four equal sides - \(\text {Side } AB = \text {Side } BC = \text {Side } CD = \text {Side } AD\)
- four right angles - \(\begin{align} \angle A = \angle B = \angle C = \angle D = \,90^\circ \end{align}\)
- two pairs of parallel sides - \(\begin{align} AB ∥ DC \qquad AD ∥ BC \end{align}\)

**Rectangle**

Properties of a rectangle

A rectangle has

- two pairs of parallel sides - \(\begin{align} AB ∥ DC \qquad AD ∥ BC \end{align}\)
- four right angles - \(\begin{align} \angle A = \angle B = \angle C = \angle D = \,90^\circ \end{align}\)
- opposite sides of equal length - \(AD = BC\) & \(AB = DC\)

**Parallelogram**

Properties of a parallelogram

- two pairs of parallel sides \(\begin{align}PQ ∥ RT \qquad PR ∥ QT \end{align}\)
- opposite sides are of equal length. \(\text{Side }PQ = \text{Side }RT\) and \(\text{Side }PR = \text{Side }QT\)

**Trapezium**

Properties of a trapezium

- One pair of parallel sides. \(\begin{align} EH ∥ GH \end{align}\)

**Rhombus**

A rhombus has

- two pairs of parallel sides. \(\begin{align} EH ∥ FG \qquad \qquad EF ∥ HG \end{align}\)
- four equal sides. \(\text{Side }EH = \text{Side }HG = \text{Side }GF = \text{Side }FE\)
- Opposite angles are equal. \(\begin{align} \angle H = \angle F = \angle E = \angle G \end{align}\)

## Angle sum property of quadrilaterals

The sum of the angles of a quadrilateral is always \(360^\circ\). This property is referred to as the **angle sum property** of a quadrilateral.

In the figure given above the sum of angles \(A\), \(B\), \(C\) & \(D\) is \(360^\circ\). This property comes handy when calculating unknown measure of angles in a quadrilateral.

## Tips and Tricks

**Tip:**While naming a quadrilateral you can name it in anti-clockwise direction. For example, in the figure given below the quadrilateral can be named as \(\square \,ABCD\) and \(\square \,ACBD\).

- While calculating the measures of an unknown angles in a quadrilateral dividing the quadrilateral into two triangles would help.

### Common mistakes or misconceptions

**Misconception 1:** Any four sided figure is a quadrilateral. For example, the figure given below is a quadrilateral.

Though the above given has 4 sides and is a closed figure, it is not a quadrilateral. Quadrilaterals are closed figures made of non-intersecting line segments.

**Misconception 2:** All rectangles are squares

Many rectangles do not have 4 equal sides and 4 equal angles. However, all squares are rectangles as they all have 4 right angles, their opposite sides are parallel to each other and opposite sides are equal to each other.

**Misconception 3:** All rhombuses are squares

Many rhombuses do not have four equal sides and four equal angles. However, all squares are rhombuses as it has 4 equal sides, its opposite sides are parallel to each other.

## Test your knowledge

Identify and name the quadrilaterals.

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