Special Parallelograms and Their Key Properties

Before beginning with the introduction of parallelogram, it is crucial to bear in mind that rectangles, squares, and rhombus (singular for rhombi) are all quadrilaterals that have all the properties of parallelograms. A substantial differentiation attribute deals with their four sides and four angles. A rectangle is a parallelogram with four right angles with two concurring sides. A rhombus, which is occasionally called a diamond, is a parallelogram with four concurring sides. And a square is a parallelogram possessing four right angles and four concurring sides.

But there’s some more! Let’s peek into each of their properties closely.

 Now, that you are well aware of the different kinds of special parallelograms. A crucial question might arise in the brilliant brains as to on what basis these quadrilaterals can be identified & how are they graded under this kind of hierarchy? There surely is something common or relatable that makes them follow such a hierarchical order.

All right, cheer up! As next up you will find the answers automatically once we complete all the properties.

So, Firstly, let’s discuss the head of hierarchy i.e. parallelogram.

Introduction of Parallelogram

As the name suggests, it must bear something parallel. So, a parallelogram is a quadrilateral with opposite sides parallel and therefore the opposite angles equal.

• Properties of a Parallelogram

Property 1: Sides opposite to each other are equal in length i.e. PQ = SR and QR = PS.

Property 2: Angles opposite to each other are equal i.e. ∠P =∠R and ∠Q = ∠S.

Property 3: The Diagonals bisect one another (at the point of their intersection) i.e. PO = RO and QO= SO

Property 4: Supplementary Consecutive angles

Now, let’s get ahead with the next in line of the hierarchy i.e. Rectangle.                       

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Introduction to Rectangle

A rectangle is a parallelogram with 2D shape in geometry and has equal angles. So, this suggests a rectangle has in affiliation all the properties of a parallelogram and more than that is having all angles equal.

• Properties of a Rectangle

Here, PQ = RS and QR = PS

And ∠P =∠Q= ∠R = ∠S   (All angles are equal)

Property 1: A rectangle is a parallelogram with ALL 4 being right angles i.e. ∠P =∠Q = ∠R = ∠S= 90°.

Property 2: Angles opposite to each other of a rectangle are equal i.e. ∠P =∠R = ∠Q= ∠S= 90°.

Property 3: The diagonals of a rectangle are congruent i.e. PR = QS

Property 4: The diagonals bisect each other (at the point of their intersection).

Property 5: Supplementary Consecutive angles

Getting to know all about the properties of a rectangle being a parallelogram, let’s get to next in heir.  

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Introduction to Rhombus:

Also known as equilateral quadrilateral, this parallelogram is with all four sides equal in length. You can easily spot a Rhombus shape object in real life. It can be found such as a kite, rhombus-shaped earring, 3D wall decals, windows of a car, and even mirrors. Also do you know what is a 3d rhombus called? It’s called Rhombohedron in geometric terms.

So, as it says a rhombus is also a parallelogram which means it has also inherited all the properties of a parallelogram and it is having all sides equal other than that.

• Properties of a Rhombus

Property 1: All four sides are equal i.e. PQ = QR = RS = PS

Property 2: Angles opposite to each other are equal.

Property 3: The Diagonals bisect each other & are perpendicular i.e. PO = RO and QO = SO and ∠POQ =∠QOR = ∠ROS = ∠SOP = 90°.

Property 4: Diagonals are NOT congruent & bisect vertices

Now, the last one left is Square.

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Introduction to Square:

A rectangle or a regular quadrilateral with sides of equal length and breadth is called a square. Seeing that the square is the endmost in the hierarchy, thus, it must have encompassed all the properties of a parallelogram, rectangle, and rhombus. Therefore, to know the properties of a square just add up all the properties you have learned until now in this lesson.

• Properties of a square

Property 1: All four sides are equal i.e. PQ = QR =RS = PS.

Property 2:  All four are right angles i.e. ∠P =∠Q= ∠R = ∠S= 90°.

Property 3:  Diagonals bisect each other & are perpendicular

Property 4:  Diagonals are all congruent & bisect vertices

Property 5:  Supplementary Consecutive angles

Solved Example 

Prove that diagonals of a Rectangle Are of Equal Length

PQRS is a rectangle with PR and QS as its diagonal

Since PQRS is a rectangle, ∠P = 90° QR=PS

QR ∥ PS and PQ is a transversal

Therefore, ∠P + ∠Q = 180°

∠Q = 90°

In (triangle) Δ PQS and Δ QPR

PQ = QP

∠P =∠Q

QR = PS

Δ PQS = Δ QPR (by law of Side Angle Side postulate)

PR= QS (Corresponding Parts of Congruent Triangle)

Hence, it proves that the diagonals of the rectangle are congruent.

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Fun Facts

1. A 1000 sided shape In geometry is called a chiliagon (a polygon with 1,000 sides)

2. The flat surfaces of many 3D geometrical figures or objects are formed of 2D shapes

3. Every rhombus is a kite, and whichever quadrilateral is both a kite and parallelogram is a rhombus

4.  A Rhombohedron is a special case of rhombus where all edges are of equal length.

There are still some questions that students have when working on this assignment. Let unlock the answers scrolling down.