Turned squares

A square piece of paper, with all sides equal and all angles 90°, is turned so that it stands on one of its corners, like a diamond. Is it still a square? Check its properties.

• Are all the sides still equal? Yes.
• Are all the angles still 90°? Yes.

Turning a square does not change its lengths or its angles, so the turned figure still has both properties of a square (S1 and S2). It is a square.

By the same reasoning, a turned rectangle is still a rectangle: its opposite sides are still equal and its angles are still 90°. A shape does not stop being a square or a rectangle just because it is tilted, just as you are still you when you tilt your head.

So, to decide whether a tilted figure is a square, do not go by how it sits on the page. Check the properties. On a dot grid, the book shows four tilted four-sided figures.

• Figure A stands on a corner, as wide as it is tall; its four sides are equal and its angles are 90°. It is a square.
• Figures B and C also have four equal sides, but B is taller than it is wide and C is wider than it is tall, and their angles are not 90°. They fail S2, so they are not squares.
• Figure D has all its angles 90°, but two of its sides are long and two are short, so it is a rectangle, not a square.

Remember: equal sides alone are not enough, and right angles alone are not enough. A square needs both.