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Lines of Symmetry in a Parallelogram

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Last updated date: 23rd Jul 2024
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An Introduction

Before we begin with the lines of symmetry of a parallelogram, we need to understand the concept of a parallelogram, its properties, its sides, angles and the corresponding relationships. A parallelogram can be defined as a special or unique kind of quadrilateral which is a closed four-sided figure with each of the opposite sides that are parallel to each other and have equal length. The interior opposite angles in any given parallelogram have equal value and any pair of adjacent interior angles present can be said as supplementary angles that is they have a sum of 180°.


What is the Symmetry of a Parallelogram?

The parallelogram has no lines of symmetry and, as with the rectangle, students should experiment with folding a copy to see what happens with the lines through the diagonals as well as horizontal and vertical lines.


For understanding the line of symmetry we need to analyse what exactly a line of symmetry is. We can say that a line of symmetry is an axis or imaginary line that can pass through the centre of a shape, facing in any direction, in such a manner that it represents mirror images of each other when cut into two equal halves for example if we cut a square or rectangle, it will have a line of symmetry because at least one imaginary line can be drawn through the centre of the shape that cuts it into two equal halves in such a manner that mirror images of each other are provided. A shape can have multiple lines of symmetry given its properties etc.


After looking at the key characteristics and other observations, it turns out that a parallelogram does not have any line of symmetry. It is a very curious question if we ask that why doesn't the parallelogram not have lines of symmetry, will the simplest answer to this question can be that it is impossible to construct a line of symmetry, an axis or an imaginary line that passes through the centre cutting its image in half where each side would represent a mirror image of the other, in order to test this you can simply try and construct a line of symmetry on any parallelogram and figure out that it is almost impossible. In addition, parallelograms do have rotational symmetry, when we see from geometry perspective of rotational symmetry refers to when a shape or figure is exactly the same as its preimage after it has been rotated a number of degrees, a parallelogram will give the same result as the original or pre-image has been rotated 180°.


After reviewing the characteristics properly and analyzing a parallelogram from all the sides we can conclude that parallelograms do not have any lines of symmetry in turn after reviewing the properties of parallelograms namely that they are quadrilaterals, we can conclude that shapes like squares and rectangles do have lines of symmetry. In the case of a parallelogram, even if they have rotational symmetry, it will result in the same exact image as we started with after a rotation of 180°.


Parallelogram and Lines of Symmetry

Have you ever wondered how many lines of symmetry a parallelogram has? A Parallelogram in general has no lines of symmetry. However, a parallelogram does have a decisive (rotational) symmetry - the half-turn around the median point at which the two diagonals intersect. As shown in the image below, because triangles WXO and YZO are congruent (have the same size and shape), thus 


WO= YO


XO = ZO


(Image will be uploaded soon)


In all Parallelograms’ Angles that are not opposite of each other will supplement up to 180 degrees. That being said, there are some specific kinds of parallelograms that have lines of symmetry. They are listed below.


Number of Symmetry Lines in Different Parallelogram


Parallelogram 

Number of Symmetry Lines 

Rhombus 

2

Rectangle 

2

Square 

4


Symmetry Lines in Different Parallelogram

  1. Lines of Symmetry in a Rhombus- 

Rhombus is a unique kind of parallelogram and it has 2 lines of symmetry - its diagonals. It means that a rhombus has reflection symmetry over either of its diagonals. Same as a parallelogram, it also has rotational symmetry of 180º about its midpoint.


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  1. Lines of Symmetry in a Rectangle -

Rectangle, which is a quadrilateral with four right angles, has 2 lines of symmetry - two lines moving through the central points of opposite sides. A rectangle has reflection symmetry when reflected over the line across the central point of its opposite sides. Same as the parallelogram, it also has rotational symmetry of 180º about its midpoint.


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  1. Lines of Symmetry in a Square

Square is a unique kind of parallelogram, with four equal sides and four equal angles. It has 4 lines of symmetry - two diagonals and two lines running through the central points of opposite sides. A square has reflection symmetry when reflected over the line across the central point of its opposite sides as well as over its diagonals. Same as the parallelogram, it also has rotational symmetry of 90º about its midpoint.


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Types of Symmetry

While you now know about how many symmetries a parallelogram has, we must also know what exactly symmetry is. Symmetry is the characteristic of being composed or created of exactly equivalent parts facing each other or around an axis. Keeping the definition of symmetry in mind, know that there are various types of symmetry in geometry. However, different types of symmetrical shapes may or may not have all or a particular type of symmetry. Thus, it is important to learn about different types of symmetrical shapes which possess or do not possess a specific type of symmetry. There are 3 types of symmetry which are as follows:-


  1. Linear Symmetry: It has1 a line of symmetry i.e. perpendicular bisector of AB

  2. Point Symmetry: It has point symmetry centre point Z of line segment AB

  3. Rotational Symmetry: It has rotational symmetry of order 2 about Z.


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Fun Facts 

  • The term "parallelogram" is derived from the Greek word "parallelogramma" (which means fenced by parallel lines).

  • Parallelograms are quadrilaterals with four sides.

  • The area is bisected by Any of the lines passing through the centre of a parallelogram.

  • A parallelogram has to its name 2 sets of parallel sides (which never meet) and four edges.

  • Opposite sides of a parallelogram are equally long (they are the same in length) and are parallel to each other.

  • Squares, Rectangles, and rhombuses are all parallelograms.

  • A trapezoid is not a parallelogram. A trapezoid has exactly two parallel sides whereas a parallelogram has two pairs of parallel sides.

  • A trapezoid is a special kind of parallelogram with at least one pair of parallel sides but neither has reflectional symmetry nor a rotational symmetry. Therefore you cannot make observations based upon symmetry.

  • An isosceles trapezoid is a parallelogram for which all four sides are the same (are equally long).

  • An isosceles trapezoid that has only one pair of parallel sides has reflectional symmetry but no rotational symmetry.

FAQs on Lines of Symmetry in a Parallelogram

1. Why there are no Lines of Symmetry in a Parallelogram?

The line of symmetry is a line that sets apart two symmetric shapes. And a parallelogram is a quadrilateral that possesses two sets of opposite sides parallel. Therefore, With respect to a parallelogram, no line of symmetry can be drawn. And though a parallelogram has a rotational symmetry when rotated at an angle of 180° about its centre, but has no reflection symmetry. Thus, no line can divide the parallelogram into two symmetric shapes.

2. A Quadrilateral has How Many Lines of Symmetry?

A quadrilateral is any geometrical figure that is a four-sided polygon. Some quadrilaterals are symmetric about their lines. Whereas, some four-sided polygons is symmetric about other diagonals. To find out the symmetry of a quadrilateral, you will use rigid-motion transformations to determine the line symmetry and rotational symmetry in different types of quadrilaterals. More so, you also might be thinking if a quadrilateral has more than two lines of symmetry? Well, a quadrilateral figure can have more than one line of symmetry subjected to the length of its sides, angles, and other characteristics. Take a regular rectangle for example- it has two lines of symmetry because it can be folded in half vertically and horizontally to create mirror images.

3. Are all Quadrilaterals Parallelograms?

A basis (non-self-intersecting) quadrilateral is also a parallelogram but if and only any one of the below statements is true.

  • Two pairs of opposite sides of a quadrilateral are equivalent in length

  • Two pairs of opposite angles are equivalent in measure

  • One pair of opposite sides are parallel and equivalent in length

  • Adjacent angles come about as supplementary

  • The diagonals happen to bisect each other 

  • Each diagonal has the quadrilateral in spits into 2 congruent triangles (like a "kite")

  • The sum total of the sides of the square is equal to the sum total of the squares of the diagonals. (as By-law of the parallelogram)

  • A quadrilateral figure has 2 lines of symmetry 

  • A figure is rotational symmetry of order 2