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

1 Answer
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 #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.