What Is Parallelogram?
A parallelogram is a special type of quadrilateral that has both pairs 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 a 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 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 On Parallelogram
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 cm2
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
Practice Problems On Parallelogram
Parallelogram
What is the perimeter of a parallelogram ABCD where the two adjacent sides are 12m and 8m, respectively?
Permieter of a parallelogram = 2 x sum of adjacent sides = 2 (12m + 8m) = 40m
Which of the following is not a parallelogram?
A parallelogram has both the pair of opposite sides as parallel and equal, whereas a trapezium has only one pair of parallel sides and no equal sides.
In a parallelogram ABCD, if ∠A = 60°, then ∠D is equal to
Opposite angles of a parallelogram are equal. So ∠A = ∠C = 60° and also ∠B = ∠D
Also, ∠A + ∠B + ∠C + ∠D = 360°
60° + ∠B + 60° + ∠D = 360°
120° + ∠B + ∠D = 360°
∠B + ∠D = 240° and ∠B = ∠D
Therefore, ∠D = 120°
Frequently Asked Questions On Parallelogram
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.
