# Why is a square a parallelogram?

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 ${90}^{\circ}$.

From the definition, it follows that a square is a rectangle. In fact, a rectangle is a quadrilateral whose interior angles measure ${90}^{\circ}$. 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 $A B C D$. The sides $A B$ and $C D$ are opposite and lie on two parallel lines. In fact, if we consider the line on which $A D$ 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 ${90}^{\circ} + {90}^{\circ} = {180}^{\circ}$. This means that the lines through $A B$ and $C D$ have to be parallel.
With the same argument one proves that $B C$ and $A D$ 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.

Jul 26, 2017

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 $4$