The properties of a Parallelogram, Rhombus, Rectangle, and Square appears to be intimidating until you don’t try to learn them. So, let’s finish this very easy topic in a very short time today and here only.
Did you know that quadrilaterals like parallelogram, rhombus, rectangle, square also have a hierarchy? Let’s make it more clear with the help of a representation.
So, now there must be a question in your mind i.e. On what basis these all quadrilaterals follow this kind of hierarchy? There must be something common or relating property which makes them follow such a hierarchal order. Don’t worry, you will find the answer to this yourself once we complete all the properties.
Firstly, let’s discuss the head i.e. parallelogram.
As the name says, it must have something parallel. So, a parallelogram is a quadrilateral which has opposite sides parallel.
Property 1: The opposite sides of a parallelogram are of equal length i.e. AB = DC and BC = AD.
Property 2: The opposite angles of a parallelogram are of equal measure i.e. ∠A =∠C and ∠B = ∠D.
Property 3: The diagonals of a parallelogram bisect each other (at the point of their intersection) i.e. AE = CE and BE = DE.
So, these were properties of a parallelogram, quite easy!
Now, let’s get to the heir of the hierarchy i.e. Rectangle.
A rectangle is a parallelogram with equal angles. So, this means a rectangle has inherited all the properties of a parallelogram and in addition to that it is having all angles equal.
Here, AB = CD and BC = AD.
And ∠A =∠B = ∠C = ∠D (All angles are equal)
Property 1: A rectangle is a parallelogram in which every angle is a right angle i.e. ∠A =∠B = ∠C = ∠D = 90°.
Property 2: The diagonals of a rectangle are of equal length i.e. AC = BD.
Property 3: The diagonals of a rectangle bisect each other (at the point of their intersection).
So, these were all properties of a rectangle being a parallelogram.
A parallelogram with sides of equal length is called a rhombus.
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.
AB = BC = CD = DA (All sides are equal)
Property 1: All sides are of equal length i.e. AB = BC = CD = DA.
Property 2: The diagonals of a rhombus are perpendicular bisectors of one another i.e. AO = CO and BO = DO and ∠AOB =∠BOC = ∠COD = ∠DOA = 90°.
Now, we are left with the last one i.e. Square.
A rectangle with sides of equal length is called a square.
Since the square is the last one in the hierarchy, therefore, it must have all the properties of a parallelogram, rectangle, and rhombus.
So, to get the properties of a square just sum up all the properties you have learned so far.
Property 1: In a square, every angle is a right angle.
Property 2: The diagonals of a square are of equal length and perpendicular bisectors of each other.
Let’s summarize all we have learned till now.
|Parallelogram||1) Opposite sides are equal.
2) Diagonals bisect one another.
3) Opposite angles are equal.
|Rectangle||1) All the properties of a parallelogram.
2) Diagonals are equal.
3) Each of the angles is a right angle.
|Rhombus||1) All the properties of a parallelogram.
2) All sides are of equal length.
3) Diagonals are perpendicular bisectors of each other.
|Square||All the properties of a parallelogram, rectangle and a rhombus.|
So, by now you must have an answer to your doubt about the hierarchal order. It is ordered on the basis of properties that we have discussed so far.
In focus: Basic Geometrical Ideas