Question

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

Answer 1

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 `color(blue)"rhombus"`
`color(blue)"All the sides are equal."`
`color(blue)"The opposite sides are parallel"`
`color(blue)"The opposite angles are equal"`
`color(blue)"The diagonals bisect each other at 90°"`
`color(blue)"The diagonals bisect the angles at the vertices"`
`color(blue)"there are 2 lines of symmetry"`
`color(blue)"it has rotational symmetry of order 2"`

A square has all the properties of a rhombus, with more properties -
In a `color(red)"square:"`
`color(blue)"All the sides are equal."`
`color(blue)"The opposite sides are parallel"`
`color(blue)"The opposite angles are equal"`
`color(red)"All the angles are equal to 90°."`
`color(blue)"The diagonals bisect each other at 90°"`
`color(red)"The diagonals are equal."`
`color(red)"The diagonals bisect the angles to give 45° angles"`
`color(red)"there are 4 lines of symmetry"`
`color(red)"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.