 # Parallelogram|Definition & Meaning

## Definition

A quadrilateral that has 2 pairs of sides that are parallel to one another is called a parallelogram. The lengths of the different sides of a parallelogram are equivalent to one another, and the opposite angles have the same degree of measurement. Additionally, the interior angles located on the same plane of the transversal are considered supplementary. The whole amount of interior angles add up to a total of 360 degrees.

## What Is a Parallelogram?

The English word “parallelogram” was taken from this Greek word. As a result, a parallelogram is defined as a quadrilateral with parallel lines serving as its boundaries. It is a shape that has opposite sides that are parallel to one another and are of equal length.

Each of the three primary forms of parallelograms—square, rectangle, and rhombus—is distinguished by a distinct set of characteristics distinct from the others.

In this article, we will study what a parallelogram is, how to calculate the area of a parallelogram, as well as other topics connected to parallelograms, followed by several instances that have been solved. Figure 1: Representation of a parallelogram

## Shapes of a Parallelogram

A parallelogram is a shape that only exists in two dimensions. It possesses a total of four sides, two pairs of which are parallel to one another. Additionally, the lengths of the sides are identical. The form in question is not a parallelogram if the lengths of the parallel sides do not measure the same. Similarly, the parallelogram’s inner angles opposite one another should always be equivalent. In such a case, we cannot call it a parallelogram.

### Special Parallelograms

Both square and rectangular: Two more shapes comparable to a parallelogram in terms of their qualities are a square and a rectangle. Both opposing sides are of the same length and run parallel. Both shapes can be cut in half along their respective diagonals.

### Rhombus

A rhombus is defined as a parallelogram in which all of the sides are congruent with one another or are equal.

### Rhomboid

A rhomboid is a specific example of a parallelogram in which the sides that are perpendicular to each other are also parallel to the sides that are adjacent to them, but the lengths of the sides that are perpendicular to each other differ. In addition, each angle adds up to precisely 90 degrees.

### Trapezium

A shape is said to be a trapezium if it has one pair of sides parallel to one another and the other two sides are not parallel to each other. Figure 2: Representation of a Trapezium.

## Angular Measurements of a Parallelogram

A parallelogram is a type of two-dimensional form that is flat and has four angles. Equal treatment is accorded to the opposing inner angles. The angles that are on the identical side of the transversal are complementary, indicating that their sum is equal to 180 degrees. Therefore, the total amount of angles that are contained within a parallelogram is equal to 360 degrees.

## Parallelogram’s Characteristics and Attributes

• A unique kind of polygon known as a parallelogram is formed when a quadrilateral has two parallel sides and faces in opposite directions. The following is a list of the characteristics of a parallelogram:
• The different sides are parallel and equivalent
• Equal to each other is the diagonal
• The angles that are successive or adjacent to one another are complementary.
• In the event that any of the angles is a right angle, then the remaining angles will also be right angles.
• Both diagonals meet and cut each other in half.
• Each diagonal cuts the parallelogram in half, creating two congruent triangles.
• A parallelogram has a unique property in which the sum of the squares of all sides is equivalent to the sum of the squares of its diagonals.

## Parallelogram’s Area

The space enclosed by all four parallelogram’s sides is referred to as the parallelogram’s area. It is possible to calculate, given the length of the parallelogram’s base and the height of the parallelogram.

Take into consideration the parallelogram RSTU with the base symbol (b) and the height symbol (h). It is possible to get the area of the parallelogram by applying the following formula: The area of a parallelogram is equivalent to the base times the height (h) Figure 3: Representation of area of a parallelogram.

## Parallelogram’s Perimeter

The length of the shape of a parallelogram is its perimeter; hence, the perimeter is equal to the total of all of the parallelogram’s sides.

Therefore, the formula for calculating the perimeter of a parallelogram with sides is:

P = 2 (a + b) units

## Example Problems Involving Parallelograms

### Example 1

A parallelogram has two adjacent angles of 4:5. Find each of its angles measurement.

### Solution

Let the parallelogram be EFGH.

Then ∠E and ∠F are the adjacent angles.

Let:

∠E = (4x) ° and ∠F = (5x) °

We know that the sum of the adjacent angles of a parallelogram is equal to 180°, so:

∠E + ∠F = 180°

4x + 5x = 180

9x = 180

x = 20

Now:

∠E = (4*20) ° = 80°

∠F = (5*20) ° = 100°

∠F + ∠G = 180° (as ∠F and ∠G are adjacent angles)

100 + ∠G = 180°

∠G = 180° – 100°

∠G = 80°

Also:

∠G + ∠H = 180°

80° + ∠H = 180°

∠H = 180° – 80°

∠H = 100°

Therefore, ∠E = 80°, ∠F = 100°, ∠G = 80°, ∠H = 100°

### Example 2

Consider a parallelogram ABCD in which ∠B = 45°. Find the measure of each of the angles ∠A, ∠C, and ∠D.

### Solution

In the given parallelogram ABCD, ∠B = 45°.

The total sum of any two adjacent angles is 180°, so:

∠A + ∠B = 180°

∠A + 45° = 180°

∠A = (180° – 45°) = 135°

Also, we know that:

∠B + ∠C = 180°

45° + ∠C = 180°

∠C = (180° – 45°) = 135°

Furthermore:

∠C + ∠D = 180° [since ∠C and ∠D are adjacent angles]

135° + ∠D = 180°

∠D = (180° – 135°) = 45°

So, ∠A = 135°, ∠C = 135° and ∠D = 45°.

All images/graphs are created using GeoGebra.

#### Parallel Lines Definition < Glossary Index > Parentheses Definition A parallelogram is indeed a fascinating geometric shape that showcases unique properties and symmetrical elegance. Defined as a quadrilateral with two pairs of parallel sides, the parallelogram offers a captivating blend of simplicity and versatility. Its distinct characteristics, such as parallel and congruent opposite sides, make it an intriguing subject of study in mathematics and geometry.

This article explores the various attributes and properties of the parallelogram, shedding light on its angles, diagonals, and relationships with other geometric shapes. Come on a journey as we explore the parallelogram’s mysteries to understand its intrinsic beauty and mathematical significance better.

## Definition

parallelogram is a geometric shape defined as a quadrilateral with opposite sides that are parallel. In other words, a parallelogram is a four-sided polygon where the opposite sides never meet and are always equidistant. This unique property sets parallelograms apart from other quadrilaterals. Additionally, opposite angles in a parallelogram are congruent, meaning they have the same measure. Parallelograms come in various forms, such as rectangles, squares, rhombi, and kites, each with its own distinguishing features. With its parallel sides and symmetrical structure, the parallelogram is a fascinating shape that finds applications in geometry, architecture, engineering, and design. Figure-1: Generic paralleogram.

## Historical Significance

The historical background of the parallelogram, a geometric shape, traces back to ancient civilizations where early mathematical principles were developed. Although it is difficult to determine the parallelogram’s precise ancestry, geometric analysis can be traced back to Mesopotamia and ancient Egypt.

In ancient Egypt, the Egyptians’ advanced knowledge of geometry was evident in their construction of monumental structures such as the pyramids. They possessed practical geometric understanding, including parallel lines and quadrilaterals concepts. The ancient Egyptians used parallelogram-like shapes in architectural designs and land surveying to ensure precise alignments.

Similarly, ancient Mesopotamia, known for its advanced mathematical knowledge, contributed to the development of geometric concepts. Babylonian mathematicians were proficient in calculating areas and volumes, which required understanding parallelograms and other geometric shapes.

The geometry study significantly progressed in ancient Greece, with notable contributions from mathematicians such as Euclid. Euclid’s work, “Elements,” written around 300 BCE, established the foundation of geometry as a formal mathematical discipline. In “Elements,” Euclid systematically presented definitions, postulates, and theorems, including those related to parallelograms. Euclid’s work shaped the study of geometry for centuries to come.

During the Renaissance, the works of mathematicians like Rene Descartes and Pierre de Fermat further expanded geometric knowledge. Their contributions included analytical geometry, which combined algebraic and geometric principles. This advancement allowed for the precise representation of geometric shapes, including parallelograms, using coordinate systems.

In modern times, the study of geometry and parallelograms has continued to evolve through the works of mathematicians, educators, and researchers. Geometry finds applications in various fields, including architecture, engineering, physics, computer graphics, etc.

## Types

### Rectangle

A rectangle is a particular kind of parallelogram with four right angles. It has all the characteristics of a parallelogram, including opposite sides that are congruent and parallel, but it also has four interior angles equal to 90 degrees. Rectangles are useful in many industries, including engineering and construction, because of their symmetry and stability, made possible by right angles. Below we present the geometric diagram of a rectangle parallelogram. Figure-2: Rectangle parallelogram.

### Square

A square is a unique type of rectangle with additional properties. It is characterized by having all sides congruent and all interior angles equal to 90 degrees. This regularity and symmetry make squares particularly appealing in both mathematics and art. Their equal sides and right angles make squares ideal for tasks that require balanced shapes and precise measurements. Below we present the geometric diagram of a square parallelogram. Figure-3: Square parallelogram.

### Rhombus

A rhombus is a parallelogram in which all four sides are congruent. Unlike rectangles or squares, rhombi typically do not have right angles. However, they share the fundamental properties of parallelograms, including opposite sides being parallel and congruent. The rhombus exhibits symmetry and aesthetic appeal, often used in geometric patterns and architectural designsBelow we present the geometric diagram of a rhombus parallelogram. Figure-4: Rhombus parallelogram.

### Kite

A kite is a special parallelogram with congruent adjacent sides. Unlike other parallelograms, kites do not have parallel opposite sides. Kites have two pairs of adjacent congruent sides and one pair of opposite congruent angles. This unique combination of side and angle properties gives kites their distinctive shape, resembling a diamondBelow we present the geometric diagram of a kite parallelogram. Figure-5: Kite parallelogram.

## Properties

parallelogram is a quadrilateral shape with several distinctive properties. Let’s explore these properties in detail:

### Opposite Sides Are Parallel

The most fundamental property of a parallelogram is that its opposite sides are parallel. This means that two sides of the parallelogram never intersect and are always equidistant.

### Opposite Sides Are Equal in Length

In addition to being parallel, the opposite sides of a parallelogram are also equal in length. This property gives the parallelogram a sense of symmetry.

### Opposite Angles Are Equal

The opposite angles of a parallelogram are congruent, meaning they have the same measure. If we label the angles of a parallelogram as A, B, C, and D in a clockwise or counterclockwise direction, then angle A is congruent to angle C, and angle B is congruent to angle D.

### Consecutive Angles Are Supplementary

The consecutive angles of a parallelogram are supplementary, meaning the sum of any two consecutive angles is equal to 180 degrees. For example, angle A + angle B = 180 degrees, and angle B + angle C = 180 degrees.

### Diagonals Bisect Each Other

The diagonals of a parallelogram are line segments that connect opposite vertices (corners) of the shape. The diagonals of a parallelogram bisect each other, meaning they intersect at their midpoints. This property divides each diagonal into two equal segments.

### Diagonals Create Congruent Triangles

When the diagonals of a parallelogram intersect, they divide the parallelogram into four triangles. The opposite triangles formed by the diagonals are congruent. This property allows for various geometric deductions and proofs.

The adjacent angles of a parallelogram, which are the angles formed by the intersection of a side and an adjacent angle, are supplementary. This means that the sum of any two adjacent angles equals 180 degrees.

### Opposite Sides and Angles Can Be Used for Congruence

In geometry, congruence refers to having the same size and shape. Parallelograms can be proven congruent based on the equality of their opposite sides and angles. They are considered congruent if two parallelograms have equal opposite sides and angles.

These properties collectively define the characteristics of a parallelogram and make it a unique and versatile geometric shape. They enable parallelogram analysis, manipulation, and application in various mathematical and practical contexts.

## Ralevent Formulas

### Area

The area of a parallelogram can be calculated using the formula A = base × height, where the height is the angle between the base and the base’s opposite side, and the base is any parallel side’s length.

### Perimeter

The perimeter of a parallelogram is the sum of all its side lengths. You can use the formula since a parallelogram’s opposite sides are congruent: P = 2 × (side1 + side2), where side1 and side2 are the lengths of any two adjacent sides.

### Diagonals

A parallelogram has two diagonals that connect opposite vertices. The length of the diagonals can be found using the formula d1 = sqrt((side1)^2 + (side2)^2 – 2 × side1 × side2 × cos(angle)), where d1 is the length of one diagonal, side1 and side2 are the lengths of adjacent sides, and angle is the measure of the angle between the adjacent sides.

### Angles

In a parallelogram, the opposite angles are congruent. The sum of the interior angles of a parallelogram is always equal to 360 degrees. The measure of each angle can be calculated using the formula angle = 180 – angle_measurement.

### Side Lengths

If you know the length of one side of a parallelogram and the measure of an adjacent angle, you can find the lengths of the other sides using the Law of Cosines. The formula is given by side2 = √(s₁²+ s₃² –  2 × s₁×s₃ × cos(θ)), where side1 and side3 are the known side lengths, and angle is the measure of the adjacent angle.

These formulas provide valuable insights into various aspects of parallelograms, such as their area, perimeter, diagonals, angles, and side lengths. They serve as powerful tools for solving problems related to parallelograms, whether it’s determining their dimensions, calculating their areas, or exploring their geometric properties.

## Applications

The parallelogram, a geometric shape, finds applications in diverse fields due to its unique properties and characteristics.

### Geometry and Mathematics

Parallelograms are a fundamental concept in geometry, forming the basis for various theorems and proofs. They are used to study and understand the properties of quadrilaterals, angles, and parallel lines. Parallelograms are also employed in coordinate geometry to represent and manipulate geometric figures.

### Architecture and Engineering

Parallelograms have practical applications in architectural and engineering design. Their properties of parallel sides and equal opposite angles allow for constructing structures with desired angles and symmetries. Parallelograms are used in floor plans, trusses, beams, and support structures, providing building stability and aesthetic appeal.

### Physics and Mechanics

Parallelograms are utilized in physics and mechanics to analyze forces and motion. The parallelogram law of vector addition states that if the adjacent sides of a parallelogram represent two vectors, then their resultant vector is given by the diagonal of the parallelogram. This principle is essential in understanding the composition of forces and their resultant effects.

### Surveying and Land Management

Parallelograms play a role in surveying and land management practices. The concept of parallel lines and parallel sides is crucial for accurately measuring and demarcating boundaries. Surveyors use parallelogram-shaped instruments, such as the theodolite, to measure angles and distances, ensuring precise land surveys.

### Art and Design

Parallelograms find application in art and design, contributing to aesthetically pleasing compositions. Artists and designers use the principles of balance, symmetry, and proportion inherent in parallelograms to create visually appealing images, patterns, and arrangements. Parallelogram-shaped elements are often incorporated into graphic design, textiles, and architecture to evoke a sense of harmony and balance.

### Computer Graphics and Modeling

Parallelograms are utilized in computer graphics and modeling to represent and manipulate two-dimensional shapes. They are employed in algorithms for rendering and transforming geometric objects. Parallelograms also serve as a basis for modeling three-dimensional shapes using techniques like extrusion and deformation.

### Optics and Light Reflection

Parallelograms have applications in optics and the study of light. In optics, the reflection of light rays follows the law of reflection, where the incident and reflected rays form equal opposite angles. This behavior is exemplified in the reflection of light from flat mirrors, which can be approximated as parallelograms.

### Pattern Recognition and Image Processing

Parallelograms are used in pattern and image processing algorithms. They are employed for shape detection, contour analysis, and feature extraction. Parallelogram-shaped structures within an image can provide valuable information for object recognition and scene understanding.

The versatile nature of parallelograms makes them valuable across various fields of study and application. Their properties and mathematical characteristics enable their use in understanding, designing, and analyzing geometric structures, physical phenomena, and visual representations.

## Exercise

### Example 2

#### Diagonals of a Parallelogram

Consider a parallelogram PQRS with side lengths PQ = 8 cm and QR = 12 cm. The diagonals PR and QS intersect at point O. Find the lengths of the diagonals.

### Solution

In a parallelogram, the diagonals bisect each other. Therefore, the lengths of the diagonals PR and QS are equal to half the sum of the lengths of the opposite sides.

Diagonal PR = (PQ + QR) / 2

Diagonal PR = (8 cm + 12 cm) / 2

Diagonal PR = 10 cm

Similarly,

Diagonal QS = (QR + RS) / 2

Diagonal QS = (12 cm + 8 cm) / 2

Diagonal QS = 10 cm

Hence, the diagonals PR and QS lengths in parallelogram PQRS are 10 cm.

### Example 3

#### The Perimeter of a Parallelogram

Considers a parallelogram LMNO as shown in Figure-6. Find the perimeter of the parallelogram. Figure-6.

### Solution

The perimeter of a parallelogram is calculated by adding the lengths of all its sides. In this case, the side lengths are LM = 5 cm, MN = LO = 9 cm, and NO = 5 cm. So, the perimeter of parallelogram LMNO is:

Perimeter = LM + MN + NO + LO

Perimeter  = 5 cm + 9 cm + 5 cm + 9 cm

Perimeter  = 28 cm

### Example 4

#### Relationship Between Sides and Angles in a Parallelogram

Consider a parallelogram XYZW with side lengths XY = 6 cm and YZ = 8 cm. If the measure of angle X is 60 degrees, find the measures of the other sides and angles.

### Solution

Since opposite sides of a parallelogram are equal, YW = XZ = 6 cm and XW = YZ = 8 cm.

To find the remaining angles, we know that opposite angles of a parallelogram are congruent. Therefore, angle Y = angle Z = 60 degrees.

Hence, the measures of the sides and angles of parallelogram XYZW are:

XY = WZ = 6 cm

YZ = XW = 8 cm

Angle X = Angle Z = 60 degrees

Angle Y = Angle W = 120 degrees

### Example 5

Consider a parallelogram ABCD with side lengths AB = 5 cm, BC = 7 cm, and angle B = 60 degrees. Find the measures of the remaining sides and angles.

### Solution

Since opposite sides of a parallelogram are equal in length, CD = AB = 5 cm. Similarly, AD = BC = 7 cm.

To find the remaining angles, we know that opposite angles of a parallelogram are congruent. Therefore, angle A = angle C = 60 degrees.

Hence, the measures of the sides and angles of parallelogram ABCD are:

AB = CD = 5 cm

BC = AD = 7 cm

Angle A = Angle C = 60 degrees

Angle B = Angle D = 120 degrees

### Example 6

In a parallelogram, if the base is 10 units long and the height is 6 units, what is the area of the parallelogram?

### Solution

Given: Base = 10 units, Height = 6 units.

The area of a parallelogram is calculated using the formula:

A = base × height

Substituting the given values, we have

A = 10 units × 6 units

A = 60 square units

Therefore, the area of the parallelogram is 60 square units.

### Example 7

In parallelogram, ABCD, the length of diagonal AC is 15 cm, and angle B = 60 degrees. Find the measures of the other sides and angles.

### Solution

Since a parallelogram’s diagonals are divided in half, AC divides parallelogram ABCD into two congruent triangles. Let’s denote the point of intersection of diagonals as E.

We can use the property of congruent triangles to find the length of DE. Since triangles ADE and CDE are congruent, we have DE = DE, meaning that DE equals half the length of AC. Therefore,

DE = AC/2

DE = 15 cm / 2

DE = 7.5 cm

Since opposite sides of a parallelogram are equal, AD = BC. Therefore, AD = BC = 7.5 cm.

To find the measure of angle A, we know that opposite angles of a parallelogram are congruent. Therefore,

angle A = angle C = 180 degrees – angle B

angle A = 180 degrees – 60 degrees

angle A = 120 degrees

Hence, the measures of the sides and angles of parallelogram ABCD are

AC = 15 cm, DE = 7.5 cm, AD = BC = 7.5 cm, Angle A = Angle C = 120 degrees, Angle B = Angle D = 60 degrees.

All images were created with GeoGebra.