# Why is a square always a rhombus, but a rhombus is not always a square?

Apr 30, 2016

It is important to work with definitions first.

#### Explanation:

A parallelogram is a quadrilateral with two pairs of opposite sides parallel.
A rhombus is a parallelogram with equal sides
A square is a rhombus with all the angles equal (to 90°).

Students often make the mistake of defining a rhombus as
"A rhombus is a square pushed over."
It would be better to say that a square is a rhombus pushed up straight.

In a $\textcolor{b l u e}{\text{rhombus}}$
$\textcolor{b l u e}{\text{All the sides are equal.}}$
$\textcolor{b l u e}{\text{The opposite sides are parallel}}$
$\textcolor{b l u e}{\text{The opposite angles are equal}}$
$\textcolor{b l u e}{\text{The diagonals bisect each other at 90°}}$
$\textcolor{b l u e}{\text{The diagonals bisect the angles at the vertices}}$
$\textcolor{b l u e}{\text{there are 2 lines of symmetry}}$
$\textcolor{b l u e}{\text{it has rotational symmetry of order 2}}$

A square has all the properties of a rhombus, with more properties -
In a $\textcolor{red}{\text{square:}}$
$\textcolor{b l u e}{\text{All the sides are equal.}}$
$\textcolor{b l u e}{\text{The opposite sides are parallel}}$
$\textcolor{b l u e}{\text{The opposite angles are equal}}$
$\textcolor{red}{\text{All the angles are equal to 90°.}}$
$\textcolor{b l u e}{\text{The diagonals bisect each other at 90°}}$
$\textcolor{red}{\text{The diagonals are equal.}}$
$\textcolor{red}{\text{The diagonals bisect the angles to give 45° angles}}$
$\textcolor{red}{\text{there are 4 lines of symmetry}}$
$\textcolor{red}{\text{it has rotational symmetry of order 4}}$

A rhombus does NOT have all the properties of a square, therefore is not a special kind of square.