It is necessary for the students to know the basics of parallelograms to practice geometry problems. Parallelogram was derived from the Greek word parallelogrammon that means bounded by parallel lines. There are three types of parallelogram they are rhombus, rectangle, square. And each type has different properties. We learn the concept of a parallelogram with all suitable examples from this page.
Parallelogram Definition
A parallelogram is a type of quadrilateral with two pairs of parallel sides. The shape of the parallelogram is a two-dimensional geometry. The opposite sides of a parallelogram are equal in length and the opposite angles of a parallelogram are equal in measures. The number of vertices in a parallelogram is four and the number of edges is also four.
Parallelogram Formula
Area of a parallelogram = height × base
The perimeter of a parallelogram = 2 × ( sum of lengths of adjacent sides)
Types of Parallelogram
A parallelogram is classified into three types. They are as follows,
1. Rectangle
2. Square
3. Rhombus
1. Rectangle:
A rectangle is a polygon with two pairs of equal sides. It has four right angles ∠A = ∠B = ∠C = ∠D = 90 degrees. Diagonals bisect each other.
2. Square:
A square is a parallelogram with all four sides and four angles are equal. AB = BC = CD = DA. Two pairs of squares have parallel sides.
3. Rhombus:
Rhombus is a type of parallelogram with all four sides and angles being the same. The diagonals of a rhombus are perpendicular to each other.
Properties of Parallelogram
Properties of the parallelogram are Used in the industry for accurate transfer of mechanical motion from one place to another.
- The opposite sides are parallel and congruent
- The opposite angles are congruent
- The consecutive angles are supplementary
- Any one of the angles is a right angle then all other angles will be at the right angle
- The two diagonals bisect each other
- Each diagonal bisects the parallelogram into two congruent triangles
- The Sum of squares of all the sides of a parallelogram is equal to the sum of squares of its diagonals, it is also called parallelogram law.
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Solved Problems on Parallelogram
Practice the problems from this page to score better marks in the exams.
Example 1.
The ratio of two sides of a parallelogram is 5: 4 if its perimeter is 90 cm. Find the lengths of its sides.
Solution:
Given
Ratio of two sides of a parallelogram is 5 : 4
Perimeter of a parallelogram is 42 cm
The length of two sides of the parallelogram be 5x cm and 4x cm
Perimeter = 2( sum of lengths of adjacent sides)
= 2( 5x + 4x ) cm
= (10x + 8x )cm
18x cm
18cm = 90
x = 90/18
x = 5
Therefore one side = 5 × 5 = 25cm
And another side = 4 × 5 = 20cm
Example 2.
In the figure PQRS is a parallelogram, PO and QO are the bisectors of ∠P and ∠Q respectively prove that ∠POQ = 90°
Solution:
We know that the sum of two adjacent angles of a parallelogram is 180°
Therefore
∠P + ∠Q = 180°……. equation 1
Here PO and QO are the bisectors of ∠P and ∠Q
∠OPQ = 1/2∠P and ∠PQO = 1/2∠Q
From ∆OPQ
The Sum of the angles of a triangle is 180°
∠OPQ + ∠POQ + ∠PQO = 180°
1/2∠P + ∠PQO + 1/2∠Q = 180°
(½ × 180°) + ∠POQ = 180° using equation 1
90° + ∠POQ = 180°
∠POQ = (180° – 90°) = 90°
∠POQ = 90°
Example 3.
In the given figure ABCD is a parallelogram in which ∠CaD = 20°, ∠BAC = 30° and ∠COD = 70° find ∠ABD
Solution:
From the figure
∠AOB = ∠COD = 70°
Now in ∆OAB we have
Sum of the angles of triangle is 180°
∠OPQ + ∠POQ + ∠PQO = 180°
30° + ∠ABO + 70° = 180°
∠ABO + 100 = 180°
∠ABO = (180° – 100°)
∠ABD = ∠ABO = 80°
Example 4.
In the figure, PQRS is a parallelogram in which ∠P = 85° find the measures of each of the angles ∠Q, ∠R, ∠S.
Solution:
Given that PQRS is a parallelogram in which ∠P = 85°
We know that
Sum of any two adjacent angles of a parallelogram is 180°
∠P + ∠Q = 180°
85° + ∠Q = 180°
∠Q = (180° – 85°) = 95°
Also
∠P + ∠Q = 95°
And
∠R + ∠S = 180°
∠85° + ∠S = 180°
∠S = (180° – 85°) = 95°
Therefore ∠Q = 95°, ∠R = 85°, and ∠S = 95°
Example 5.
Calculate the area of ground in the shape of a parallelogram, the base measures 30 in and the altitude measures 7 in.
Solution:
Given that
Base = 30 in
Altitude = 7in
We know that,
Area of parallelogram = b × h
Area of ground = 30in × 7in = 210 in²
Thus the area of the ground is 210 sq. in.
FAQs on Concept of Parallelogram
1. How many sides does a parallelogram have?
A parallelogram has four sides.
2. What are the 4 properties of a parallelogram?
The 4 properties of a parallelogram are
i. The opposite sides are parallel.
ii. The diagonals of a parallelogram bisect each other.
iii. Opposite sides are congruent.
iv. Consecutive angles are supplementary
3. Do parallelograms have right angles?
If one of the angles is a right angle, then all four angles must be right angles.