# Why is a square a parallelogram?

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See explanation.

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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

From the definition, it follows that a square is a rectangle. In fact, a **rectangle** is a quadrilateral whose interior angles measure

Let's show (the more general fact) that rectangles are parallelograms.

Consider a rectangle

With the same argument one proves that

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.

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

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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

2 pairs of opposite angles are equal

The diagonals bisect each other

It has rotational symmetry of order

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

All its angles are equal (to

The diagonals bisect each other at

The diagonals are equal.

The diagonals bisect the angles at the vertices to give

It has rotational symmetry of order

Describe your changes (optional) 200