Lines of Symmetry in a Parallelogram

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

In all Parallelograms’ Angles that are not opposite of each other will supplement up to 180 degrees.

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That being said, there are some specific kinds of parallelogram 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 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 symmetry does a parallelogram have, 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 does not possess a specific type of symmetry. There are 3 types of symmetry which are as follows:-

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

  2. Point Symmetry: It has point symmetry center 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 line 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 parallelogram are equally long (they are 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 a 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 same (are equally long).

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

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FAQ (Frequently Asked Questions)

Why there are no Lines of Symmetry in a Parallelogram?

The line of symmetry is a line which 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 1800 about its center, but has no reflection symmetry. Thus, no line can divide the parallelogram in two symmetric shapes.

A Quadrilateral has How Many Lines of Symmetry?

A quadrilateral is any geometrical figure that is four-sided polygon. Some quadrilaterals are symmetric about their lines. Whereas, some four-sided polygon are 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.

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 happens 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 parallelogram)

  • A quadrilateral figure has 2 lines of symmetry 

  • A figure is rotational symmetry of order 2