A parallelogram is one of the typical shapes you often come across in geometry class. Although it looks like a slanted rectangle, a parallelogram has some special properties that make it unique.
With knowledge of parallelogram shapes, types, and properties, you can easily relate them to real-life objects and solve mathematical problems involving similar shapes. This blog offers useful insights into parallelograms to make understanding this four-sided shape simple and fun.
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A parallelogram is a two-dimensional geometrical shape with four sides; the opposite pair of sides are parallel and equal in length. For better visualization, you can consider a parallelogram with some unique properties, like a slanted rectangle. Understanding a parallelogram helps us recognize how this shape forms the basis of many mathematical and real-world applications.
A parallelogram is a quadrilateral in which the pairs of opposite sides are parallel and equal in length.
For example, ABCD is a parallelogram.
Therefore, we can say,
AB and CD are parallel and equal
AD and BC are parallel and equal
A parallelogram has its own set of unique properties. To truly understand what a parallelogram is, it’s essential to learn the properties of a parallelogram. These properties hold for all types of parallelograms and can be used to prove or solve various geometric problems.
Key Properties:
Opposite sides are equal and parallel
Opposite angles are equal
Adjacent angles are supplementary (sum is 180°)
The sum of all angles is 360°
Diagonals bisect each other
Each diagonal divides the parallelogram into two congruent triangles
Let’s explain these properties with the help of the following diagram.
In the above parallelogram ABCD, we can relate the following properties:
Opposite sides are parallel, which means AB || DC and AD || BC
Opposite sides are equal, which means AB = DC and AD = BC
Opposite angles are equal, which means ∠A = ∠C and ∠B = ∠D
Adjacent angles are supplementary which means ∠D + ∠C = 180° and ∠A + ∠B = 180°
The sum of all angles is 360° which means ∠A + ∠B + ∠C + ∠D = 360°
Diagonals bisect each other, which means AE = CE and DE = BE
Each diagonal divides the parallelogram into two congruent triangles, which means that triangles ABD and CBD are congruent and triangles ADC and ABC are congruent.
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The definition of a parallelogram indicates that there can be different parallelogram shapes that satisfy the properties of a parallelogram. There are several types of parallelograms, each with distinct characteristics, as mentioned below.
Rectangle
It is a type of parallelogram in which all the internal angles are 90 degrees, and opposite sides are equal and parallel. The diagonals are of equal length.
In the rectangle ABCD, we can relate the following properties:
Two pairs of parallel sides: AB || DC and AD || BC
Four right angles: ∠A = ∠B = ∠C = ∠D = 90∘
Opposite sides of equal lengths: AB = DC and AD = BC
Two equal diagonals: AC = BD
Diagonals bisect each other.
Rhombus
It is a type of parallelogram in which all sides are equal and opposite sides are parallel. The internal angles are not right angles. The diagonals are equal in length and bisect each other at right angles.
In the rhombus ABCD, we can relate the following properties:
Two pairs of parallel sides: AB || DC and AD || BC
Four equal sides: AB = BC = CD = DA
Equal opposite angles: ∠A = ∠C and ∠B = ∠D
Diagonals that are perpendicular to each other: BD⊥AC
Diagonals bisect each other: BM = DM and AM = CM
Square
This type of parallelogram has four equal sides, and all internal angles are 90 degrees. The opposite pair of sides is parallel. The diagonals are equal and bisect each other at right angles.
In the square ABCD, we can relate the following properties:
Two pairs of parallel sides: AB || DC and AD || BC
Four equal sides: AB = BC = CD = DA
Four right angles: ∠A = ∠B = ∠C = ∠D = 90∘
Diagonals are equal: AC = BD
Diagonals are perpendicular to each other: BD⊥AC
Diagonals bisect each other: AO = CO and DO = BO
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General Parallelogram
It is the typical parallelogram with no right angles, and opposite sides are equal and parallel. The diagonals may or may not be equal in length, but they bisect each other.
In the general parallelogram ABCD, we can relate the following properties:
Two pairs of parallel sides: AB || DC and AD || BC
Opposite sides are equal: AB = DC and AD = BC
Equal opposite angles: ∠A = ∠C and ∠B = ∠D
Diagonals bisect each other: AO = CO and DO = BO
After getting an idea about the types and properties of parallelograms, knowing the correct formulas becomes important for solving problems involving area and perimeter. The formulas for different parameters of a parallelogram are explained as follows:
The area of a parallelogram is calculated by multiplying the length of its base and its height (the perpendicular distance between the base and the opposite parallel side).
So, the formula for the area of a parallelogram is:
Area (A) = Base (b) X Height (h)
The perimeter of a parallelogram is the total length of all four sides. The formula for the perimeter of a parallelogram is:
Perimeter (P) = 2 (sum of adjacent sides)
= 2 (a + b)
Where a and b are the lengths of adjacent sides.
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1. Find the area and perimeter of a parallelogram with base 15 cm, height 8 cm, and side length 6 cm.
Solution:
Area = base × height = 15 × 8 = 120 cm²
Perimeter = 2 × (15 + 6) = 2 × 21 = 42 cm
2. In a parallelogram ABCD, the sum of the two opposite angles ∠A and ∠C is 70 degrees. Find all four angles of the parallelogram.
Solution:
In the parallelogram, the opposite angles are equal.
Therefore, ∠A = ∠C
Again, ∠A + ∠C = 70
Or, ∠A + ∠A = 70
Or, 2∠A = 70
Or, ∠A = 35
Therefore, ∠C = 35 degrees
Again, the sum of adjacent angles in the parallelogram is 180 degrees.
So, ∠A + ∠D = 180 degrees
Or, ∠D = 180 - ∠A
Or, ∠D = 180 – 35 = 145
Again, in the parallelogram, ∠D = ∠B
So, we can say ∠B = 145 degrees
Therefore, the measurements of all four angles of the parallelogram are:
∠A = 35 degrees, ∠B = 145 degrees, ∠C = 35 degrees, and ∠D = 145 degrees
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Studying parallelograms and their properties helps strengthen your understanding of geometry, vectors, and algebra. In higher mathematics, concepts from parallelograms apply to different topics, which include:
Coordinate geometry
Trigonometry
Vector mathematics
Physics and engineering problems
Understanding the parallelogram as a special geometrical shape clarifies mathematical concepts and real-world applications. Knowing how to use parallelogram formulas, recognise properties, and identify types of parallelograms helps in problem-solving in competitive exams quickly and accurately.
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