Special Parallelograms

Rectangles, rhombuses and squares belong to the set of parallelograms. Each of these may be defined as a parallelogram as follows:

• A rectangle is an equiangular parallelogram.

• A rhombus is an equilateral parallelogram.

• A square is an equilateral and equiangular parallelogram.

Thus a square is both a rectangle and a rhombus.

The relations among the special parallelograms can be pictorially represented in the figure given below:

Since every rectangle and every rhombus must be a parallelogram, they are shown as subsets of a parallelogram and since a square is both a rectangle and rhombus, it is represented by the overlapping shaded section. Let us now define each of these special types of parallelograms.

Rectangle

A rectangle is a parallelogram with one of its angle a right angle.

Note:

It can be shown that each angle of a rectangle is a right angle.

(sum of interior angles on the same side of transversal AB)

Corollary:

Each of the four angles of a rectangle is a right angle.

Rhombus

A rhombus is a parallelogram with a pair of its consecutive sides equal.

ABCD is a rhombus in which AB=BC.

Since a rhombus is a parallelogram, AB=DC and BC=AD.

Thus AB=BC=CD=AD.

Area of Rhombus can be given by A = 1/2 * d1* d2, where d1 and d2 are diagonals of Rohmbus AC and BD respectively.

Corollary:

All the four sides of a rhombus are equal (congruent).

Square

A square is a rectangle with a pair of its consecutive sides equal.

Since square is a rectangle, each angle of a rectangle is a right angle and AB=DC, BC=CD.

Thus AB=BC=CD=AD.

Each of the four angles of a square is a right angle and each of the four sides is of the same length.