Angle Sum Property of a Quadrilateral

Chapter -8

Quadrilaterals

A quadrilateral has four sides, four angles and four vertices [see Fig. (i)].

In quadrilateral ABCD, AB, BC, CD and DA are the four sides; A, B, C and D are the four vertices 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 [see Fig (ii)]. AC and BD are the two diagonals of the quadrilateral ABCD.
In this chapter, we will study more about different types of quadrilaterals, their properties and especially those of parallelograms.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 (see Fig).

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 rectangular 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 (see Fig).
What is the sum of angles in Δ ADC?

You know that
 

Similarly, in

Δ ABC,

    

Adding (1) and (2), we get

   

   

       

 i.e., the sum of the angles of a quadrilateral is 360^o.
Look at the different quadrilaterals drawn below

 

Observe that:
One pair of opposite sides of quadrilateral ABCD is Fig. (i) namely, AB and CD parallel. You know that it is called a trapezium.
Both pairs of opposite sides of quadrilaterals given in Fig. (ii), (iii), (iv) and (v) are parallel. Recall that such quadrilaterals are called parallelograms.
So, quadrilateral PQRS of Fig. (ii) is a parallelogram.