Question

Why is a square a parallelogram?

Answer 1

See explanation.

Explanation:

A parallelogram is a quadrilateral with two pairs of opposite sides.
A square is a quadrilateral whose sides have equal length and whose interior angles measure `90^@`.

From the definition, it follows that a square is a rectangle. In fact, a rectangle is a quadrilateral whose interior angles measure `90^@`. This is one of the two conditions expressed above for a quadrilateral to be a square, so a square is also a rectangle.

Let's show (the more general fact) that rectangles are parallelograms.
Consider a rectangle `ABCD`. The sides `AB` and `CD` are opposite and lie on two parallel lines. In fact, if we consider the line on which `AD` lies, this is a transverse of the pair of lines. The internal angles in `A` and in `D` are alternate interior angles, and the sum of their measures is `90^@+90^@=180^@`. This means that the lines through `AB` and `CD` have to be parallel.
With the same argument one proves that `BC` and `AD` lie on parallel lines, and this proves that every rectangle is a parallelogram.

Another (maybe longer) way of proving this fact is to use the condition on the sides of a square (i.e. that all the sides have equal length) and observe that a square is also a rhombus. Then, by showing that every rhombus is a parallelogram, you found another way of proving that every square is a parallelogram.

Answer 2

A square has all the properties of a parallelogram and can therefore be considered to be a parallelogram

Explanation:

The properties of a parallelogram can be stated according to:

• the sides
• the angles
• the diagonals
• the symmetry

A parallelogram has:

2 pairs of opposite sides parallel
2 pairs of opposite sides equal

The sum of the angles is `360°`
2 pairs of opposite angles are equal

The diagonals bisect each other

It has rotational symmetry of order `2`

All of these properties apply to square, so it can be considered to be a parallelogram.

However a square has additional properties as well, so it can be regarded as a special type of parallelogram.

A square has:

2 pairs of opposite sides parallel
All its sides equal

The sum of the angles is `360°`
All its angles are equal (to `90°)`

The diagonals bisect each other at `90°`
The diagonals are equal.
The diagonals bisect the angles at the vertices to give `45°` angles.

`4` lines of symmetry
It has rotational symmetry of order `4`