Determine whether the following statement is true or false: a square is a rectangle.
Let’s begin this question by recalling the mathematical definition of a square. And that is that it’s a four-sided polygon or quadrilateral with all sides equal and all interior angles are 90 degrees. Let’s see if we can draw out some different squares. We could draw two different squares that look like this. The side lengths in each square are equal until the interior angles are 90 degrees. We could draw a very tiny square or even a much larger square. In each of our drawings, each of the side lengths within the square are equal and the interior angles are 90 degrees.
So how does this compare with a rectangle? We can define a rectangle as a four-sided polygon or quadrilateral, where all interior angles are 90 degrees. So let’s have a look at our squares and see if they fit the definition of a rectangle. In the first square that we’ve drawn, we can see that all the interior angles are 90 degrees, so this first square is also a rectangle. In our second shape, once again, we do have all interior angles of 90 degrees.
And the same is true for each of the last two squares. Both of these are also rectangles. This is because when we drew our squares, one of the things we had to make sure was that all the interior angles were 90 degrees. This is what we need to have in a four-sided polygon in order to make it a rectangle. So therefore, we would say that the statement that a square is a rectangle is true because every square that we draw will also have interior angles of 90 degrees, which means that it would be a rectangle.
If we wanted to think of this more diagrammatically, we could think of a Venn diagram. The region of squares sits within the region of rectangles. We could interpret this by saying that every square is a rectangle, but not every rectangle is a square.