Determine whether the following statement is true or false: a rectangle is a square.
It might be useful here to begin by recalling the mathematical definition of a rectangle. A rectangle is a four-sided polygon, or quadrilateral, where all interior angles are 90 degrees. So, we could draw a rectangle that looks like this. It’s got a length and width of six and four, but all the interior angles are 90 degrees. We could draw a different rectangle that looks like this, like this, or even like this. The important thing is that each of these rectangles that we’ve drawn have interior angles of 90 degrees.
So, now, let’s look at the definition of a square. This is a four-sided polygon with all sides equal and all interior angles are 90 degrees. So, let’s look at the rectangles we’ve drawn and see if any of these fit to the definition of a square. In this first example, we do not have all sides equal, even though all the interior angles are 90 degrees. So, this would not be a square. In the second diagram, we do not have all sides equal. So, this one is not a square, neither is the third diagram that we’ve drawn. In the last diagram, however, we do have a square because we have all the sides equal and all interior angles are 90 degrees.
So, to answer the question “is a rectangle a square,” we would have to say no, as there’s only one time when this occurs, whenever all the sides are equal. Because we found a number of examples where a rectangle is not a square, then we have to give the answer that this is false. To further illustrate this, if we were to draw a Venn diagram of rectangles and squares, then the region of squares would fit within the rectangles. We could interpret this by saying that all squares are rectangles, but not all rectangles are squares.