Special Parallelograms: Rhombus, Square & Rectangle

Parallelogram is a quadrilateral with two pairs of parallel sides in two dimensions. A parallelogram's opposite angles and opposite sides are congruent, and the total of its internal angles is 360°. Rhombus, Square and Rectangle are all types of Parallelograms which possess all the properties of a parallelogram and also have their own unique properties. Thus they are also called Special Parallelograms.

Key Terms: Parallelograms, Special parallelogram, Square, Rectangle, Rhombus

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Parallelograms

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"Parallelogram" is derived from the Greek word "parallelogrammon," which means "bounded by parallel lines." As a result, a parallelogram is a quadrilateral that is divided by parallel lines. It is a shape with opposite sides that are parallel and equal.

Parallelogram

There are six key properties of parallelograms to be aware of:

1. Opposite sides are congruent (AB = DC).
2. Opposite angels are congruent (D = B).
3. Consecutive angles are supplementary (A + D = 180°).
4. If one angle is right, then all angles are right.
5. The diagonals of a parallelogram bisect each other.
6. A parallelogram is divided into two congruent triangles by each diagonal.

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Special Parallelograms

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A rectangle, a square and a rhombus are special parallelograms because they not only show the properties of a parallelogram but also have unique properties of their own. 

Rectangle

A quadrilateral having four right angles is known as a rectangle. As a result, every angle in a rectangle is the same (360°/4 = 90°). Also, the diagonals of a rectangle bisect each other, and the opposite sides of a rectangle are parallel and equal. Some of the real-life examples of a rectangle are books, mobile phones, etc.

Rectangle

Properties of Rectangle:

• The opposite sides are equal in length.
• The opposite sides are parallel.
• The interior angles measure 90 degrees each.
• Diagonals are equal in length.
• Diagonals bisect each other.
• Each diagonal bisects the rectangle into 2 congruent triangles.
• It has 2 lines of symmetry – a horizontal and a vertical.
• The mid-points of the 4 sides of a rectangle, when joined in order, form a rhombus.

Rectangle formula:

If the length of the rectangle is L and breadth is B, then,

• Area of a rectangle = Length × Breadth or L × B
• Perimeter of rectangle = 2 × (L + B)

Square

A quadrilateral having four equal sides and angles, is known as a square. It is also a regular quadrilateral, with equal sides and angles. A square, like a rectangle, has four 90-degree angles. It can alternatively be visualized as a rectangle with two equal sides. Some of the real-life examples of a square are a slice of bread, a chessboard etc.

Square

Properties of Square:

• All 4 sides are equal in length.
• The opposite sides are parallel.
• The interior angles measure 90 degrees each.
• Diagonals are equal in length.
• Diagonals bisect each other at right angles.
• Diagonals bisect opposite vertex angles.
• It is a highly symmetric figure with 4 lines of symmetry – a horizontal, a vertical, and 2 diagonals.
• Each of the diagonals bisects the square into 2 congruent triangles. In fact, both the pairs of congruent triangles are also congruent to one another.

Square formula

If the side of a square is ‘a’ then,

• Area of the square = a × a = a²
• Perimeter of the square = 2 × (a + a) = 4a

Rhombus

A quadrilateral with equal-length sides and opposite sides that are parallel to each other is known as a rhombus. The angles aren't all 90 degrees, though. A rhombus with right angles would make a square. Because it resembles the diamond suit in playing cards, the rhombus is frequently referred to as the 'diamond.'

Rhombus

Properties of a Rhombus

• Opposite angles are equal.
• All sides are equal and, opposite sides are parallel to each other.
• Diagonals bisect each other perpendicularly, at right angles.
• The sum of any two adjacent angles is 180°
• Diagonals bisect opposite vertex angles.
• Each of the diagonals divides the rhombus into 2 congruent triangles.

Rhombus Formulas

• If the side of a rhombus is a, then, the perimeter of a rhombus = 4a.
• If the length of the two diagonals of the rhombus is d1 and d2, then the area of a rhombus = ½ × d1 × d2.

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Comparison of the Properties

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Comparison of the properties of all the special parallelograms is tabulated below.

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Things to Remember

• A quadrilateral is a four-sided two-dimensional figure with 360 degrees of inner angles.
• A quadrilateral with four sides, two of which are parallel to each other, is known as a parallelogram.
• A square is a specific parallelogram with diagonals that are perpendicular to each other and are both equilateral and equiangular.
• A rectangle is a particular parallelogram with congruent opposite sides and 90° angles on each side.
• A rhombus, often known as a diamond, is a unique parallelogram with four congruent sides and diagonals that are perpendicular to one another.
• Every square is a rectangle.
• Every square is a rhombus.
• Every rhombus, square and rectangle is a parallelogram.
• All parallelograms are quadrilaterals.

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