Properties of parallelogram and its Types
A parallelogram is a special kind of quadrilateral in which both pairs of opposite sides are parallel. In fig. 1 given below, AD||BC and AB||CD .
The four basic properties of parallelogram are:
- Opposite sides of a parallelogram are equal
- Opposite angles of parallelogram are equal
- Diagonals divide the parallelogram into two congruent triangles
- Diagonals bisect each other
There are three special types of parallelogram, they are:
Let us discuss these special parallelograms one by one.
Rectangle is a special case of parallelogram in which measure of each interior angle is \( 90^\circ \) . It is an equiangular quadrilateral.By equiangular quadrilateral, it means that all the interior angles are equal in magnitude. Since the measure of each angle is \( 90^\circ \), therefore sum of its opposite angles is supplementary and hence it is a cyclic quadrilateral i.e. all its vertices lie on circumference of a circle.
The path which surrounds a two-dimensional object is known as its perimeter. The two-dimensional space occupied by an object is known as its area. A rectangle with length l units and breadth as b units has perimeter equal to 2( l + b) units and its area is equal to \( l \times b \) sq. units.
Fig. 2 shown above represents a rectangle in which all angles are right angles and opposite sides are equal.
Since a rectangle is a parallelogram, it inherits all the properties of parallelogram along with some special properties which differentiates it from other parallelograms:
Properties of rectangle:
- Measure of each interior angle is \( 90^\circ \)
- Opposite sides are equal
- Diagonals are congruent
- Each diagonal is angle bisector of opposite angle
A parallelogram in which all four sides are equal in length is known as a rhombus. A rhombus is an equilateral quadrilateral. By equilateral quadrilateral, we mean a quadrilateral with all sides equal. Every rhombus is a parallelogram since it has both pairs of opposite sides parallel. When all the sides of a kite become equal in length, then that kite becomes a rhombus as sides are of equal length and diagonals are perpendicular to each other. Hence, every rhombus is also a kite.
Fig. 3 represents a rhombus with sides AB = BC = CD = DA and diagonals intersecting at right angles.
Since a rhombus is a parallelogram, all the properties of a parallelogram are applicable to it.
Properties of Rhombus:
- All sides are congruent
- Diagonals bisect each other at right angles
- Opposite angles are equal and each diagonal is angle bisector of opposite angle
Geometrically, polygons are related with their duals i.e. vertices of one polygon correspond to sides of another polygon. Rhombus is the dual polygon of rectangle and vice- versa. By joining mid points of sides of rectangle a rhombus is formed and vice-versa. This could be understood through the table given below:
|Equiangular quadrilateral- All angles are equal||Equilateral quadrilateral- All sides are equal|
|Opposite sides are congruent||Opposite angles are congruent|
|Point of intersection of diagonals is equidistant from vertices therefore it circumscribes a circle.||Point of intersection of diagonals is equidistant from sides therefore it inscribes a circle.|
|Length of diagonals is equal||Intersection of diagonals is at equal angles|
|Opposite sides are bisected by its axis of symmetry||Opposite angles are bisected by its axis of symmetry|
Table1: Duality of Rectangle and Rhombus
Square: A parallelogram which has properties of both a rhombus and a rectangle. A square is a parallelogram with equal sides and one of its interior angles as right angle.
Fig. 4 depicts a square in which measure of each interior angle is \( 90^\circ \) and AB = BC = CD = DA.
Properties of Square:
- All sides are equal in length
- Each interior angle is right angle
- Length of diagonals is equal
- Diagonals are perpendicular bisectors of each other
- Every square is a parallelogram in which diagonals are congruent and bisect the angles
- Every square is a rectangle and a rhombus
Here are different types of Parallelogram and their properties. For more on parallelograms watch the video.