## What is Parallelogram?

A parallelogram is a special type of quadrilateral that has both the pair of opposite sides parallel and equal.

The given figure shows a parallelogram ABCD which has AB II CD and AD II BC. Also, AD = BC and AB = CD.

**Non-Example:**

A trapezium is the non-example of a parallelogram.

Real-Life Examples of a Parallelogram

When we look around us, we can see multiple parallelogram-like shapes and objects in the form of buildings, tiles, or paper.

**Buildings**: Many buildings are constructed, keeping in mind the shape of parallelograms. A famous real-life illustration is the Dockland Office Building in Hamburg, Germany.

**Tiles**: Tiles come in various shapes and sizes. One of the most found tile shapes is that of a parallelogram.

**Eraser**: Everyone is familiar with the classic eraser. Erasers, too, come in several shapes and sizes, one of them being that of a parallelogram. The faces of this eraser are in the shape of a parallelogram.

Properties of Parallelograms

- In a parallelogram, the opposite sides are parallel to each other. Here, AB || CD, and AC || BD.
- The opposite sides of a parallelogram are equal in length. Here, AB = CD and AC = BD
- The measurement of opposite angles of a parallelogram is equal. Here, ∠A = ∠C and ∠B = ∠D
- Like all other quadrilaterals, the sum of all the angles of a parallelogram is 360°.
- The adjacent or the adjoining angles of a parallelogram add up to 180°. Therefore, ∠A + ∠B = 180°, ∠B + ∠C = 180°, ∠C + ∠D = 180°, and ∠D + ∠A = 180°.
- The diagonals of a parallelogram bisect each other. Here, OB = OD, and OA = OC.
- The diagonals AC and BD in the figure divide the parallelogram into two congruent triangles.

Types of Parallelograms

There are** **three unique kinds of parallelograms:

- Rhombus: A rhombus is a parallelogram in which all sides are equal. Here, AB = BC = CD = DA. ABCD is a rhombus.

- Square: A square is a parallelogram where all sides and diagonals are equal. The angles are right angles. Here, AB = BC = CD = DA and ∠A = ∠B =∠C =

∠D = 90 degrees and also AD = BC. ABCD is a square.

- Rectangle: A rectangle is a parallelogram in which all angles are 90°, and the diagonals are equal. The opposite sides have equal lengths. Here all angles are right angles. Diagonals PN and OM are equal. MNOP is a rectangle.

## Area of a Parallelogram

The area of a parallelogram is given by the formula** A = bh**, where b is the length of the base, and “h” is the height.

## The Perimeter of a Parallelogram

The perimeter of a parallelogram equals the sum of the lengths of the four sides. Since the opposite sides of a parallelogram are equal, its perimeter can also be expressed as 2 x the sum of adjacent sides, i.e., 2 (AB + BC)

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## Solved Examples

### Example 1.

In the figure below, ABCD is a parallelogram where ∠DAB = 75° and ∠CBD = 60°. Calculate ∠BDC.

**Solution:**

As we know, the opposite angles of a parallelogram are equal. Therefore, ∠DCB = ∠DAB = 75°.

We also know that the sum of the angles of a triangle is 180°. Now, consider ∆ BCD. Here, ∠BDC + ∠DCB + ∠CBD = 180°

We know that ∠DCB = ∠DAB = 75°. Therefore:

∠BDC + ∠DCB + ∠CBD = 180°

⇒ ∠BDC + 75° + 60° = 180°

⇒ ∠BDC + 135° = 180°

⇒ ∠BDC = 180° – 135° = 45°

Therefore, ∠BDC = 45°

### Example 2

Find the area of this parallelogram with a base of 15 centimeters and a height of 6 centimeters.

**Solution:**

A = b × h

A = (15 cm) × (6 cm)

A = 90 cm^{2}

**Example 3**

Two adjacent sides of a parallelogram are 5 cm and 3 cm. Find its perimeter.

**Solution:**

We know that opposite sides of a parallelogram are equal.

Suppose we have a parallelogram ABCD, then:

AB = CD = 5 cm and

BC = AD = 3 cm

Perimeter of parallelogram = 2 (AB + BC) = 2 (5 + 3) cm

= 16 cm

**Frequently Asked Questions**

**Is a trapezium a parallelogram?**

No, a trapezium is not a parallelogram because there are two pairs of parallel sides in a parallelogram, whereas a trapezium has only one pair of parallel sides.

**Are all rectangles parallelograms?**

Yes, all rectangles are parallelograms because a rectangle has two sets of parallel sides and two pairs of opposite sides that are equal. Therefore, it abides by all the properties of a parallelogram.

**Are all parallelograms rectangles?**

No, a parallelogram is not always a rectangle. In a rectangle, all angles must be equal to 90°. But for a parallelogram, no angles need to be equal to 90°.

**Do diagonals of a parallelogram always bisect each other?**

Yes, diagonals of a parallelogram always bisect each other.

**Is it necessary that all sides be equal in a quadrilateral to be a parallelogram?**

No, a parallelogram does not have all sides equal. Only opposite sides of a parallelogram are equal.