Geometry - Parallelogram
- Table of Contents
- What is a Parallelogram
- Properties of Parallelogram
- Cyclic Parallelogram
- Types of Parallelogram
- Square
- Rectangles
- Rhombus
- Formulas Related to Parallelogram
- Solved Examples on Parallelogram
- Practice Problems
- Frequently Asked Questions
Introduction
A parallelogram is a quadrilateral with two sets of parallel lines. The pair of parallel sides are equal in length. Like any other `2D` shape, a parallelogram too has a set of properties that defines its shape and size. We will learn about various characteristics, formulas and other information related to parallelograms.
What is a Parallelogram?
To understand a parallelogram, we must first understand what a quadrilateral is. A quadrilateral is a form of polygon with four sides and the sum of internal angles equal to `360^\circ` degrees.
The Latin words 'Quadra', which means four, and 'Latus', which means 'sides', are combined to form the word ‘quadrilateral’. A quadrilateral is a closed shape with four sides, four vertices, and four angles. We can have several quadrilaterals depending on the sides and angles. One of these quadrilaterals is a parallelogram.
A quadrilateral with opposite sides equal in length that are parallel to each other is known as a parallelogram. Now, let us look at the figure `ABCD` given below, which is a parallelogram in which `AB∥CD` and `AD∥BC`. The opposite sides are of the same length. The parallelogram has four edges and four vertices.
Properties of Parallelogram
A parallelogram is a unique kind of quadrilateral. Some of the properties of the parallelogram `PQRS` are stated below:
- The parallelogram's opposite sides are parallel. The opposite sides `PQ` and `SR`, and `PS` and `QR`, are parallel to one another.
- The opposite sides of the parallelogram are also congruent. The opposite sides `PQ` and `SR`, and `PS` and `QR`, are congruent to one another.
- The opposite angles of the parallelogram are also congruent. The opposite angles, i.e., `∠SPQ` and `∠QRS`, and `∠PQR` and `∠PSR` are congruent to one another.
- Interior angles of the same sides are supplementary to one another. Here, the interior angles of the identical sides are `∠SPQ` and `∠PQR, ∠PQR` and `∠QRS, ∠QRS` and `∠RSP,` and `∠RSP` and `∠SPQ`. If these interior angles are summed in pairs, the result will always be equal to `180^\circ`.
- The diagonal of a parallelogram divides it into two congruent triangles. The diagonals, `PR` or `QS`, divide the parallelogram into two different congruent triangles, namely `\trianglePSQ` and `\triangleQSR` or `\trianglePQR` and `\trianglePSR`.
- The diagonals `PR` and `QS`, bisects each other.
Cyclic Parallelogram
If all four vertices of a quadrilateral are located on a circle, it is cyclic. The term "inscribed quadrilateral" is also used to refer to it. In geometry, the circumcircle or circumscribed circle of a polygon is the circle that passes through each of the polygon's vertices. The sum of the opposite angles here is supplementary.
Let the four angles of the given inscribed parallelogram be `∠T, ∠P, ∠Q,` and `∠S`.
The sum of the opposite angles of a cyclic parallelogram, `∠T + ∠Q = 180^\circ` and `∠P + ∠S = 180^\circ`.
Types of Parallelogram
There can be several types of parallelograms such as:
- squares
- rectangles
- rhombus
Square
A parallelogram with four equal sides and equally sized angles is called a square. All the angles of a square are right angles, i.e. all angles are `90^\circ`. The diagonals of the square are also equal and bisect at a `90^\circ` angle.
A square can also be described as a rectangle with two opposite sides that are of the same length. Additionally, it is true to say that a quadrilateral is a square if it is both a rectangle and a rhombus.
Rectangles
A rectangle is a type of parallelogram in which opposite sides are equal and parallel to each other, similar to that of a parallelogram. Particularly, a parallelogram with adjoining sides that differ in length and one whose angles are all right angles. A rectangle has two dimensions, i.e., length and width because it is a two-dimensional form.
Rhombus
Rhombus is a special parallelogram that has its opposite sides parallel to each other, but adjacent sides are of unequal lengths. Also, the opposite angles are equal. And the angles are non-right angles. Every rhombus is considered to be a parallelogram, while the opposite is not always true.
Formulas Related to Parallelogram:
The formulas related to the parallelogram are listed below:
Area of Parallelogram`=` base `×` height
Perimeter of Parallelogram`= 2`(base `+` side)
Solved Examples on Parallelogram
Example `1`: Find the area of the parallelogram if the base and height are `20` cm and `5` cm respectively.
Solution:
From the given question, base `= 20` cm and height `= 5` cm.
Substitute the values in the formula, Area `=` base `×` height
Area `= (``20`` × ``5``)` cm square
Area `= 100` cm square
Therefore, the area of the parallelogram is `100` cm square.
Example `2`: Find the perimeter of the parallelogram if the base and side are `10` m and `2` m respectively.
Solution:
From the given question, base `= 10` m and height `= 2` m.
Substitute the values in the formula, Perimeter `= 2`(base `+` side)
Perimeter `= 2(``10`` + ``2``)` m
Perimeter `= 24` m
Therefore, the perimeter of the parallelogram is `24` m.
Practice Problems
Q`1`. What is the perimeter of a parallelogram `ABCD` with adjacent sides of `15` m and `25` m?
- `40` m
- `80` m
- `55` m
- `60` m
Answer: b
Q`2`. Which one of the following is not a parallelogram?
- Rhombus
- Rectangle
- Square
- Trapezium
Answer: d
Q`3`. If `∠B = 80°` in a parallelogram `ABCD`, then `∠A` is equal to -
- `200°`
- `150°`
- `100°`
- `170°`
Answer: c
Frequently Asked Questions
Q`1`. What are the basic characteristics of a parallelogram?
Answer: The basic characteristics of a parallelogram are:
- The sides that are opposite must be parallel to one another.
- The opposite angles must be congruent.
- The opposite sides must be congruent.
Q`2`. What makes parallelograms unique?
Answer: A quadrilateral with two sets of parallel sides is referred to as a parallelogram. The lengths of the opposing sides are equal and parallel. The opposite angles have equal measures. It features two diagonals that cut over each other, and the adjacent angles are supplementary.