Which Property Makes A Rectangle, A Special Type of Parallelogram?

The classification of shapes and figures collected and compiled by Euclid while teaching mathematics at Alexandria, Egypt, in 300 BCE, in a famous volume called ‘Elements’ gave rise to the principles of Euclidean Geometry.

Our understanding of the classification of shapes and figures is influenced by Euclidean Geometry to a large extent.

In order to understand which property makes a rectangle a special type of parallelogram, let us start by understanding the properties of parallelograms and how parallelograms are a subset of quadrilaterals.

What is a quadrilateral?

The word quadrilateral is a combination of two Latin words ‘Quadri’, meaning four, and ‘latus’, meaning side, respectively. As per Euclidean Geometry, a quadrilateral is defined as a two-dimensional figure with four sides, whose sum of internal angles is equal to 360°. Quadrilaterals have distinct properties that distinguish them from other polygons.

So, what are the distinct properties of quadrilaterals? 

Properties of a quadrilateral

There are two identifying properties of quadrilaterals that differentiate them from other geometrical shapes, such as circles and triangles:

• A quadrilateral is a closed shape with four sides or line segments that meet at endpoints or vertices.
• The sum of all the internal angles of a quadrilateral is equal to 360°.

However, the above properties imply that the sides of a quadrilateral can be equal or unequal, which brings us to the next concept.

There can be different types of quadrilaterals based on their shapes, such as a parallelogram or a trapezium.

What is a parallelogram?

As the name suggests, a parallelogram is simply a quadrilateral with two pairs of parallel sides or line segments. Additionally, the opposite angles in a parallelogram are equal, and its diagonals bisect each other.

Properties of a parallelogram

A quadrilateral must satisfy certain properties to be classified as a parallelogram. A parallelogram has four identifying properties that distinguish it:

• It is a closed shape with four sides or line segments in which the opposite sides are equal and parallel.
• The angles opposite each other in a parallelogram are equal.
• The diagonals bisect each other.
• The sum of any two adjacent angles is equal to 180°.

That’s why a parallelogram is considered different from a trapezium, which has only one pair of parallel sides.

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Area and perimeter of a parallelogram

The two dimensions of a parallelogram are length and breadth. If the length of a parallelogram is ‘l,’ breadth is ‘b’ and height is ‘h’ then:

Perimeter of parallelogram= 2 × (l + b)

Area of the parallelogram = l × h

Different types of a parallelogram

Parallelograms can be further divided into various types based on their special characteristics. Some prominent examples are as under:

• Rhombus: A rhombus is a parallelogram in which all sides are equal, opposite sides are parallel to each other, and opposite angles are equal.
• Rectangle: A rectangle is a parallelogram in which the opposite sides are equal and parallel to each other, and all angles are equal to 90°.
• Square: A square is a parallelogram in which all sides are equal, opposite sides are parallel to each other, and all angles are equal to 90°.

Let’s examine which property makes a rectangle a special type of parallelogram.

For this, we first need to understand the properties of a rectangle.

Since a rectangle is a type of parallelogram, the opposite sides of a rectangle are parallel and equal, and its diagonals bisect each other. Additionally, all the angles in a rectangle are equal to 90° (360°Ã·4). Thus, a rectangle has four right angles.

Properties of a rectangle

A rectangle has three important properties:

• It is a closed shape with four sides or line segments in which the opposite sides are equal and parallel.
• All the angles of a rectangle are 90°.
• The diagonals are equal and bisect each other.

Area and perimeter of a rectangle

The two dimensions of a rectangle are length and breadth. If the length of a rectangle is ‘l’ and the breadth is ‘b,’ then:

Perimeter of the rectangle = 2 × (l + b) 

Area of the rectangle = l × b. A rectangle is considered a special case of a parallelogram because of the following reasons:

• We now know that a parallelogram should have two pairs of opposite sides that are equal and parallel.
• A rectangle is a parallelogram with two pairs of opposite sides that are equal and parallel. Also, a rectangle has right angles between adjacent sides.

Thus, for a parallelogram to be a rectangle, it is necessary to fulfill the condition of each angle of the parallelogram to be 90°. This is because this factor does not hold true for all parallelograms. A parallelogram doesn’t need to have any of the angles at 90°, e.g., a rhombus. 

The relation between quadrilaterals, parallelograms, and rectangles

We can understand the relation between quadrilaterals, parallelograms, and rectangles in the following manner.

• Quadrilaterals are closed polygons that have four sides.
• Parallelograms are those quadrilaterals in which opposite sides are parallel and equal in length.
• Rectangles are parallelograms in which all angles are 90°.

This implies that rectangles are a special subset of parallelograms, which in turn are a subset of quadrilaterals.

Are all rectangles a type of parallelogram?

Yes, all rectangles are a special type of parallelogram. This is because they satisfy parallelograms’ condition that opposite sides should be parallel and equal in length.

Are all parallelograms a type of rectangle?

No, all parallelograms are not a type of rectangle. A rectangle has properties similar to a parallelogram, but only a parallelogram with special properties of equal opposite sides and all angles at 90° is called a rectangle.

As we have seen earlier, even a rhombus is a parallelogram in which all sides are equal, opposite sides are parallel to each other, and opposite angles are equal. But the angles are not at 90°, which leads to its separate classification.

How is a rectangle different from a rhombus?

A rhombus is also a type of parallelogram with four sides. However, it has the following characteristics:

• Opposite sides are equal and parallel to each other.
• Opposite angles are equal.
• The angles need not be at 90°.
• All sides are equal.

Hence, a rectangle is different from a rhombus because it does not meet the last two conditions of a rhombus, which is also a parallelogram. In a rectangle, the opposite sides are equal and parallel rather than all sides being equal. Also, a rectangle requires all angles to be at 90°, a condition that is not satisfied by a rhombus.

How is a rectangle different from a square?

A square is also a type of parallelogram with four sides. However, it has the following characteristics:

• Opposite sides are equal and parallel to each other.
• Opposite angles are equal.
• All angles are at 90°.
• All sides are equal.

Hence, a rectangle is different from a square because it does not meet the last condition of a square, though it satisfies the first three conditions. In a rectangle, the opposite sides are equal and parallel rather than all sides being equal.

Area and perimeter of a square

If the side of a square is called ‘a,’ then:

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

Area of the square = a × a = a²

Are all squares a special type of rectangle?

Yes, all squares are a special type of rectangle. This is because they satisfy the condition for rectangles having right angles between adjacent sides, in addition to two pairs of opposite sides that are equal and parallel. So, they can be considered as special rectangles where all sides are equal.

Are all rectangles a type of square?

No, the reverse does not hold true. A square is a parallelogram with properties similar to a rectangle. Still, only a parallelogram that possesses special properties of all sides being equal and all angles at 90° are called a square. As we know, a rectangle is a parallelogram in which the opposite sides are equal and parallel, but all sides need not be equal.

Summary:

To summarize, we must remember that a rectangle is a special type of parallelogram with four right angles. Similarly, a square can be considered a special case of both a rectangle and a parallelogram, with all angles at 90° and all sides being equal. To know more check out the website.

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