Why is a square a parallelogram?

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52
Nov 15, 2015

Answer:

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.

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13
Jul 26, 2017

Answer:

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#

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