## What are Quadrilaterals?

Quadrilaterals are one type of polygon which has four sides and four vertices and four angles along with 2 diagonals. There are various types of quadrilaterals.

## Types of Quadrilaterals

The classification of quadrilaterals are dependent on the nature of sides or angles of a quadrilateral and they are as follows:

- Trapezium
- Kite
- Parallelogram
- Square
- Rectangle
- Rhombus

### Trapezium

A trapezium is a quadrilateral with a **pair of parallel** sides.

Parallelogram

A **parallelogram** is a quadrilateral whose opposite sides are **parallel **and **equal**.

Rhombus

- A
**rhombus**is a quadrilateral with**sides**of**equal length**. - Since the
**opposite sides**of a rhombus have the**same length**, it is also a**parallelogram**. - The
**diagonals**of a**rhombus**areof one another.**perpendicular bisectors**

## Revisiting Geometry

### Introduction to Curves

A** curve** is a geometrical figure obtained when a** number of points **are joined without** lifting** the pencil from the paper and** without retracing** any portion. It is basically a **line** which** needÂ not be straight**.

The various types of curves are:

- Open curve: An
**open curve**is a curve in which there is**no path**from any of its point to the**same point**. - Closed curve: A
**closed curve**is a curve that forms**a path**from any of its point to the**same point**.

A curve can be :

- A closed curve

- Â anÂ open curve

- A closed curve which is not simple

### Polygons

A simple **closed curve **made up of only** line segments** is called a **polygon**.

Various examples of polygons are Squares, Rectangles, Pentagons etc.

Note:

The sides of a polygon do not cross each other.

For example,Â the figure given below is not a polygon because its sides cross each other.

### Classification of Polygons on the Basis of Number of Sides / Vertices

Polygons are classified according to the number of sides they have. The following lists the different types of polygons based on the number of sides they have:

- When there are three sides, it isÂ
**triangle** - When there are four sides, it isÂ
**quadrilateral** - When there are fives sides, it isÂ
**pentagon** - When there are six sides, it isÂ
**hexagon** - When there are seven sides, it isÂ
**heptagon** - When there areÂ eight sides, it is
**octagon** - When there are nine sides, it isÂ
**nonagon** - When there are ten sides, it isÂ
**decagon**

### Diagonals

A **diagonal** is a line segment connecting two **non-consecutive vertices** of a **polygon**.

### Polygons on the Basis of Shape

Polygons can be classified as **concave** or **convex **based on their shape.

- A
**concave**polygon is a polygon in which at leastÂ one of its**interior angles**is**greater than 90**âˆ˜. Polygons that are**concave**have at least**some****portions of their diagonals**in their**exterior**. - A
**convex**polygon is a polygon with all its**interior angle****less than 180**âˆ˜.Â Polygons that are**convex**have**no portions**of their**diagonals**in their**exterior**.

### Polygons on the Basis of Regularity

Polygons can also be classified as **regularÂ polygons** and** irregular polygons**Â on the basis of regularity.

- When a polygon is both
**equilateral**and**equiangular**itÂ is called as a regular polygon. In a regularÂ polygon, all the sides and all the angles are equal. Example: Square - AÂ polygon which is not regular i.e. it is not equilateral and equiangular, is an irregular polygon. Example: Rectangle

## Introduction to Quadrilaterals

### Angle Sum Property of a Polygon

According to the** angle sum property** of a polygon, the **sum of all the interior angles** of a polygon is equal to (nâˆ’2)Ã—180âˆ˜, where *nÂ *is the number of sides of the polygon.

As we can see for the above quadrilateral, if we join one of the diagonals of the quadrilateral, we get two triangles.

The sum of all the interior angles of the twoÂ triangles is equal to the sum of all the interior angles of the quadrilateral, which is equal to 360âˆ˜ = (4âˆ’2)Ã—180âˆ˜.

So, if there is a polygon which has** nÂ sides**, we can make

**(**which will perfectly cover that polygon.

*n*– 2) non-overlapping trianglesThe **sum of the interior angles of the polygon** will be equal to the **sum of the interior angles of the triangles** = (nâˆ’2)Ã—180âˆ˜

### Sum of Measures of Exterior Angles of a Polygon

The **sum** of the measures of the **external angles** of any** polygon** is **360**âˆ˜.

## Properties of Parallelograms

### Elements of a Parallelogram

- There are
**four sides**and**four angles**in a parallelogram. - The
**opposite sides**and**opposite angles**of a parallelogram are**equal**. - In the parallelogram ABCD, the sides Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯AB and Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯CD are
**opposite**sides and the sidesÂ Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯AB andÂ Â¯Â¯Â¯Â¯Â¯Â¯Â¯Â¯BC are**adjacent sides**. - Similarly, âˆ ABC and âˆ ADC are
**opposite angles**and Â âˆ ABC and âˆ BCD are**adjacent angles**.

### Angles of a Parallelogram

The **opposite angles **of a parallelogram are **equal**.

In the parallelogram ABCD, âˆ ABC=âˆ ADC and âˆ DAB=âˆ BCD.

TheÂ **adjacent angles **in a parallelogram are** supplementary**.

âˆ´ In the parallelogram ABCD, âˆ ABC+âˆ BCD=âˆ ADC+âˆ DAB=180âˆ˜

### Diagonals of a Parallelogram

The **diagonals** of a parallelogram** bisect **each other at the point of intersection.

In the parallelogram ABCD given below, OA = OC and OB = OD.

## Properties of Special Parallelograms

### Rectangle

A **rectangle** is a **parallelogram** with **equal angles** and each angle is** equal to 90**âˆ˜.

Properties:

**Opposite sides**of a rectangle are**parallel**andÂ**equal**.- The length of
**diagonals**of a rectangle is**equal**. - All the
**interior angles**of a rectangle are**equal to 90**âˆ˜. - The
**diagonals**of a rectangle**bisect**each other at the point of intersection.

### Square

A **square **is a** rectangle** with **equal sides**. All the properties of a rectangle are also true for a square.

In a square the diagonals:

- bisect one another
- are of equal length
- are perpendicular to one another