# Quadrilaterals and their properties

## Introduction to Quadrilaterals and their properties

If we take four straight lines, and then join their ends progressively one after the other, in order to form a closed figure, what we get is a quadrilateral. Outside the triangle and the circle, the quadrilateral is the most common shape we encounter in geometry. Quadrilaterals have some unique properties, and they also have a few commonly recognizable types, all of which we would be discussing today. For that, we need to understand the basic common features of all quadrilaterals.

## The Big Idea: Quadrilaterals and their properties

### A simple idea: The standard features of any quadrilateral

Before we begin to understand the properties of quadrilaterals, we must understand the basic features of quadrilaterals. The first is the simplest, the four sides we discussed at the start. These are straight lines which define the areas or space bounded by the quadrilateral. Next, we see that these four sides form a closed figure in which there are four internal angles. Finally, if we extend all four sides of the quadrilateral towards both sides, we will see that each side and the extension of its adjacent side enclose an angle which is referred to as the exterior angle of a quadrilateral, and there are four such angles.

### Types of Quadrilaterals

We know quadrilaterals by different names, depending on specific characteristics they have. These are the six special types of quadrilaterals possible:

**Rectangle:**A quadrilateral with opposite sides equal and also parallel, and all four internal angles are equal to 90°.**Square:**All 4 sides are equal to each other and all 4 internal angles are equal to 90°. The opposite sides are parallel.**Parallelogram:**Opposite sides are equal and parallel to each other. You can say that a rectangle is a specific type of parallelogram and a square is a specific type of rectangle.**Rhombus:**All 4 sides are equal, and opposite sides are parallel.**Trapezium:**One pair of sides is parallel to each other. If the non-parallel sides are equal to each other, then the trapezium is called a regular trapezium.**Kite:**Both pairs of adjacent (__not parallel, please note__) sides are equal.

### Properties of Parallelograms

- A quadrilateral with opposite sides equal as well as congruent is called a parallelogram.
- The opposite angles of a parallelogram are also congruent.
- The adjacent angles add up to 180°.
- The diagonals of a parallelogram always bisect each other.
- Both the diagonals also divide the parallelogram into two congruent triangles.

### Properties of Rectangles

- If all the four internal angles of a parallelogram are right angles (90°), then it is called a rectangle.
- The opposite sides are parallel and congruent.
- If we denote the length of a rectangle by L and its breadth by B, then the area of the rectangle is the product of L and B, while its perimeter is equal to L + B + L + B = 2L + 2B = 2(L+B).
- In order to find the length of the diagonal of a rectangle, you can use the Pythagoras Theorem easily, because all 4 internal angles of a rectangle are right angles. So, if D is the length of a diagonal, then D
^{2}= L^{2}+ B^{2}.

### Properties of Squares

A square is a special type of rectangle in which all sides and all internal angles are equal.

The 4 internal angles of a square are equal to 90°.

The 4 sides of a square are equal to each other.

A square is a special type of rectangle; therefore, we can say that a square is a special kind of parallelogram with all sides and angles equal.

So, we represent all sides of a square by the same letter, says, and the area of a square = L x B = s x s = s^{2}. The perimeter of a square = 2(L + B) = 2 (s + s) = 2(2s) = 4s.

### Properties of Rhombus

- A rhombus is a special type of quadrilateral in which all sides and the opposite angles are congruent.
- In a rhombus, the diagonals bisect each other at right angles.
- All four sets of adjacent angles in a rhombus are supplementary, which means they always add up to 180°.
- Therefore, we use the diagonal lengths of a rhombus to find its area. If a and b are the lengths of the two diagonals, then the area of the rhombus is \(\frac{(a\times b)}2\).

### Properties of Trapezium

- A trapezium is a quadrilateral in which one set of opposite sides is parallel to each other.
- If the lengths of the two parallel sides are represented by A and B, and the perpendicular distance between these two parallel sides is represented by h, then the area of the trapezium is calculated as: \({1 \over 2}\) \({ \times \text{(A+B)} \times h}\).