Objective:

This topic gives an overview of; 

• Quadrilaterals
• Angle Sum Property of a Quadrilateral
• Types of Quadrilaterals

Introduction

You have studied many properties of a triangle in previous chapters and on joining three non-collinear points in pairs, the figure so obtained is a triangle. Now, let us mark four points and see what we obtain on joining them in pairs in some order.

Note that if all the points are collinear (in the same line), we obtain a line segment, if three out of four points are collinear, we get a triangle, and if no three points out of four are collinear, we obtain a closed figure with four sides.

Such a figure formed by joining four points in an order is called a quadrilateral. In this book, we will consider only quadrilaterals of the type given.

A quadrilateral has four sides, four angles and four vertices.

In quadrilateral ABCD, AB, BC, CD and DA are the four sides; A, B, C and D are the four vertices and and ∠ A, ∠ B, ∠ C and ∠ D are the four angles formed at the vertices.

Now join the opposite vertices A to C and B to D. AC and BD are the two diagonals of the quadrilateral ABCD. AC and BD are the two diagonals of the quadrilateral ABCD.

You may wonder why should we study about quadrilaterals (or parallelograms) Look around you and you will find so many objects which are of the shape of a quadrilateral - the floor, walls, ceiling, windows of your classroom, the blackboard, each face of the duster, each page of your book, the top of your study table etc. Some of these are given below.  

Although most of the objects we see around are of the shape of special quadrilateral called rectangle, we shall study more about quadrilaterals and especially parallelograms because a rectangle is also a parallelogram and all properties of a parallelogram are true for a rectangle as well.

Angle Sum Property of a Quadrilateral

Let us now recall the angle sum property of a quadrilateral. The sum of the angles of a quadrilateral is 360º. This can be verified by drawing a diagonal and dividing the quadrilateral into two triangles.

Let ABCD be a quadrilateral and AC be a diagonal. What is the sum of angles in Δ ADC?

You know that

∠ DAC + ∠ ACD + ∠ D = 180°----(i)

Similarly, in ∆ ABC, ∠ CAB + ∠ ACB + ∠ B = 180°----(ii)

Adding (1) and (2), we get

∠ DAC + ∠ ACD + ∠ D + ∠ CAB + ∠ ACB + ∠ B = 180° + 180° = 360°

Also, ∠ DAC + ∠ CAB = ∠ A and ∠ ACD + ∠ ACB = ∠ C

So, ∠ A + ∠ D + ∠ B + ∠ C = 360°

i.e., the sum of the angles of a quadrilateral is 360°

Types of Quadrilaterals

Look at the different quadrilaterals drawn below:

Observe that :

• One pair of opposite sides of quadrilateral ABCD  namely, AB and CD are parallel. You know that it is called a trapezium.
• Both pairs of opposite sides of quadrilaterals given are parallel. Recall that such quadrilaterals are called parallelograms.                                                                                     

So, quadrilateral PQRS  is a parallelogram.Similarly, all quadrilaterals given above are parallelograms.

• In parallelogram MNRS, note that one of its angles namely ∠ M is a right angle.It is called a rectangle.
• The parallelogram DEFG has all sides equal and we know that it is called a rhombus.
• The parallelogram ABCD  has ∠ A = 90° and all sides equal; it is called a square.
• In quadrilateral ABCD of  AD = CD and AB = CB i.e., two pairs of adjacent sides are equal. It is not a parallelogram. It is called a kite.

Note that a square, rectangle and rhombus are all parallelograms.

• A square is a rectangle and also a rhombus.
• A parallelogram is a trapezium.
• A kite is not a parallelogram.
• A trapezium is not a parallelogram (as only one pair of opposite sides is parallel in a trapezium and we require both pairs to be parallel in a parallelogram).
• A rectangle or a rhombus is not a square.

 We have a rectangle and a parallelogram with same perimeter 14 cm.

Here the area of the parallelogram is DP × AB and this is less than the area of the rectangle, i.e., AB × AD as DP < AD. Generally sweet shopkeepers cut ‘Burfis’ in the shape of a parallelogram to accomodate more pieces in the same tray.

Summary:

• Sum of the angles of a quadrilateral is 360°.
•  A diagonal of a parallelogram divides it into two congruent triangles.
• A quadrilateral is a parallelogram, if
  • opposite sides are equal or
  •  opposite angles are equal
  •  diagonals bisect each other
  • a pair of opposite sides is equal and parallel
• Diagonals of a rectangle bisect each other and are equal and vice-versa.
• Diagonals of a rhombus bisect each other at right angles and vice-versa.
• Diagonals of a square bisect each other at right angles and are equal, and vice-versa.