Why is a square a parallelogram?

2 Answers

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^@90.

From the definition, it follows that a square is a rectangle. In fact, a rectangle is a quadrilateral whose interior angles measure 90^@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 ABCDABCD. The sides ABAB and CDCD are opposite and lie on two parallel lines. In fact, if we consider the line on which ADAD lies, this is a transverse of the pair of lines. The internal angles in AA and in DD are alternate interior angles, and the sum of their measures is 90^@+90^@=180^@90+90=180. This means that the lines through ABAB and CDCD have to be parallel.
With the same argument one proves that BCBC and ADAD 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