What is a Parallelogram Definition Properties Formula and Examples
In two dimensions, a parallelogram is a geometric shape with parallel sides. It has two parallel sides that are of the same length, making it a four-sided polygon (sometimes referred to as a quadrilateral). The sum of the adjacent angles of a parallelogram is 180 degrees.
Parallelograms are a unique class of polygons. It is a quadrilateral in which the opposite sides of both pairs are parallel. A parallelogram's characteristics make it simple and quick to distinguish between several given shapes.
The word "parallelogram," which means "bounded by parallel lines," is derived from the Greek word "parallelogramma". Let's find out more about parallelograms, their characteristics, and some examples.
Concept of a Parallelogram
Having two sets of parallel sides makes a quadrilateral a parallelogram.
A parallelogram has opposite sides that are the same length and angles that are of the same measure.
Additionally, the internal angles on the transverse side are supplementary. The total internal angles add up to $360^\circ $.
A parallelepiped is a three-dimensional shape with faces that are parallelograms.
The area of a parallelogram is given by its base, which is one of its parallel sides, and height, which is the difference between its top and bottom heights. The lengths of a parallelogram's four sides determine the perimeter of that parallelogram.
Parallelogram ABCD with Equal Diagonals
As you can see from the previous figure, ABCD is a parallelogram where $AB\parallel CD$ and $AD\parallel BC$.
Furthermore, $AB = CD$ and $AD = BC$ .
In addition, $\angle A = \angle C$ & $\angle B = \angle D$ .
$\angle A$ and $\angle D$ are supplementary angles as well because they are located on the same side of the transversal. The additional angles $\angle B$ and $\angle C$ function similarly.
Consequently, $\angle A + \angle D = 180^\circ $, $\angle B + \angle C = 180^\circ $.
Shape of a Parallelogram
A parallelogram has two dimensions. Each of its four sides has two parallel pairs of sides. Additionally, the parallel sides have the same length. If the parallel sides are not equal, the shape is not a parallelogram. Similarly to this, a parallelogram must always have equal inner angles on both sides. Otherwise, the given shape will not be a parallelogram.
Some Special Types of Parallelograms
Square and Rectangle: Two forms that resemble a parallelogram in terms of its characteristics are the square and the rectangle. Both have their opposing sides parallel to and equal to one another. Both squares and rectangles have diagonals that bisect each other in two equal halves.
Rhombus: If all of a parallelogram's sides are congruent or equal to one another, the parallelogram is a rhombus.
Rhomboid: This is a unique type of parallelogram with opposite sides that are parallel to one another but the neighbouring sides are of different lengths. The angles are also all 90 degrees.
Trapezium: If there are two parallel sides and two non-parallel sides, the shape will be a trapezium.
Different Types of Parallelograms
What Are the Properties of a Parallelogram?
A quadrilateral is a special type of polygon, which can be known as a parallelogram if it contains two parallel opposite sides. A parallelogram has the following properties:
Properties of a Parallelogram
The opposing sides are congruent and parallel. Segment AB is parallel and congruent to segment DC, while segment AD is parallel and congruent to segment BC.
The opposing angles show congruency. Angle A is congruent with angle C, while angle D is congruent with angle B.
Consecutive interior angles on the same side have a supplementary sum. Angles A and D are supplementary, as angles B and C. Similarly, angles A and B, angles D and C.
All of the angles will be at right angles if any one of them is a right angle.
Each of the two diagonals intersects the other diagonal.
A parallelogram can be divided into two congruent triangles by each of its diagonals. Triangle DCB and triangle DAB are congruent.
Formulas for Area and Perimeter of a Parallelogram
The next section discusses the formula for a parallelogram's area and perimeter. These equations can be used by students to solve difficulties.
Parallelogram's Area
A parallelogram's area is the area that it takes up in a two-dimensional plane. The formula to calculate the parallelogram area is listed below:
Base x Height = Area
Parallelogram
Parallelogram's Perimeter
The sum of the distances between the parallelogram's boundaries equals its perimeter. We must be aware of the length and width measurements in order to determine the perimeter value. The parallelogram's opposing sides are of equal length. In order to compute the perimeter, the following formula is used:
Perimeter $ = 2(a + b)$ units.
where the parallelogram's side lengths are $a$ and $b$.
Solved Examples
Q.1. The quadrilateral ABCD has an AB value of $8.5$ units. ABCD's diagonals are right angles to one another. Determine ABCD's perimeter.
Solution. A quadrilateral is a rhombus if its diagonals intersect at right angles, as determined by the characteristics of parallelograms.
Therefore, ABCD is a rhombus, and $AB = BC = CD = DA$ follows.
The perimeter then equals $4 \times AB = 4 \times 8.5 = 34$ units.
Q.2. Determine the area of a parallelogram with a $5$cm base and an $8$ cm height.
Solution. The base is $5$ cm, and the height is $8$ cm.
Knowing that Area = Base x Height
Area $ = 5 \times 8$
Area $ = 40$sq. cm
So, the area is $40$sq.cm
Practice Questions
A parallelogram with sides of $20$ and $15$ metres has a diagonal of length $15$ metres. What is the area of such a parallelogram?
Find the area of a parallelogram whose adjacent sides are $15$ cm and $8$ cm.
Answers
$100\sqrt 5 $ sq. m
$46$ sq. cm
Summary
The learning outcomes from the parallelogram topic will help in properly understanding all the geometric shapes related to a parallelogram, how to determine a parallelogram's area and perimeter, and examples of where we might see a parallelogram in daily life. In this article, we have undertaken an in-depth study of the geometrical figure, parallelogram and also discussed its properties. Do try to solve the examples and practice questions provided above to enhance your understanding of the topic further.
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FAQs on Parallelogram Concepts Definition and Applications
1. What is a parallelogram in geometry?
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel and equal in length. In geometry, it has the following properties:
- Opposite sides are parallel and equal.
- Opposite angles are equal.
- Adjacent angles are supplementary (sum to 180°).
- Diagonals bisect each other.
2. What is the formula for the area of a parallelogram?
The area of a parallelogram is given by the formula Area = base × height (A = b × h). Here:
- b is the base (any side of the parallelogram).
- h is the perpendicular height from that base.
3. How do you find the perimeter of a parallelogram?
The perimeter of a parallelogram is calculated using P = 2(a + b), where a and b are the lengths of adjacent sides. Steps:
- Measure one pair of adjacent sides.
- Add them together.
- Multiply the sum by 2.
4. What are the properties of a parallelogram?
The main properties of a parallelogram describe its sides, angles, and diagonals. These include:
- Opposite sides are equal and parallel.
- Opposite angles are equal.
- Consecutive angles are supplementary (sum = 180°).
- Diagonals bisect each other.
5. How do you find the height of a parallelogram?
The height of a parallelogram can be found using the formula h = Area ÷ base. Steps:
- Write the area formula A = b × h.
- Rearrange to h = A ÷ b.
- Substitute the known values.
6. What is the difference between a parallelogram and a rectangle?
The main difference is that a rectangle has four right angles, while a parallelogram does not necessarily have right angles. Comparison:
- Both have opposite sides equal and parallel.
- A rectangle has all angles = 90°.
- A parallelogram can have angles other than 90°.
7. Do the diagonals of a parallelogram bisect each other?
Yes, the diagonals of a parallelogram bisect each other, meaning they cut each other into two equal parts. If the diagonals intersect at point O:
- AO = OC
- BO = OD
8. Can a parallelogram have equal diagonals?
A parallelogram has equal diagonals only if it is a rectangle or a square. In a general parallelogram:
- Diagonals bisect each other.
- But they are not necessarily equal.
9. How do you prove a quadrilateral is a parallelogram?
A quadrilateral is a parallelogram if it satisfies at least one key condition. You can prove it by showing:
- Both pairs of opposite sides are parallel.
- Both pairs of opposite sides are equal.
- Diagonals bisect each other.
- One pair of opposite sides is equal and parallel.
10. What are real-life examples of a parallelogram?
Real-life examples of a parallelogram include shapes where opposite sides are parallel and equal. Common examples are:
- A slanted book or notebook shape.
- Parallelogram-shaped floor tiles.
- Designs in bridges and mechanical linkages.
