# Why is a square a parallelogram?

See 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.