Name: Jaireh
Who is asking: Student
Level: Secondary

This is something that aroused a debate in class: A rectangle was defined as a parallelogram with 4 right angles. A square was defined as a parallelogram with 4 congruent sides and 4 right angles.

I need written and conclusive proof that some rectangles can or cannot be squares. I tried insisting that some of them can.. but without proof nobody will listen.

Hi Jaireh,

There are a couple of different approaches to such a 'proof' becasue there are couple of differnt issues about what makes people 'listen', 'be convinced', ...

Logically. Look at your definitions.
Take a square

a parallelogram with 4 congruent sides and 4 right angles. Logically (AND), you can drop one of the qualities and the other still holds: a parallelogram with .... 4 right angles. Therefore it satisfies the definition of a square.

Take a program like Geometers Sketchpad. Construct a 'rectangle' - by taking two sides with a right angle (but different lentgths). Then take 'parallel line' constructions to get the other two sides (making a parallelogram with four right angles). NOTE that this actually shows that a parallelogram with ONE right angle is a rectangle (because all the other angles automatically fall into place!).

Now use the dynamic properties of the program to adjust, with the mouse, the lenght of one of the sides. As it moves, you move among a lot of figures, all of which are rectangles (by construction). In ONE of these, the two sides you started with happen to be equal in length. This is the square, a special case, but still a rectangle. Heck when one side is double the other it is a special case (for some folks) but it is still a rectangle. Or when one side is a special fraction of the other (called the 'Golden Ratio') it is a special case called the Golden Rectangle. Still a rectangle.

If you do not have access to this program, you can download a demo version of it and do the construction live (but can't save). Alternately, you can download Cinderella ( which is more complicated, but lets you save a construction with up to 18 objects (points, lines, ... ).

Of course, you can do the same with objects - like sticks etc. and slide things around for people to see. The advantage of the program is: it is more accurate and less work!

By the way, there is a general phrase for this kind of 'definition'. The definition of a rectangle is 'inclusive'. If you make a Venn diagram of squares, rectangles etc., the set of squares is 'included' in the set of rectangles. Mathematicians make a habit of prefering inclusive definitions. They are easier to write, to think about, to use.

Why would you WANT to say a square is NOT a rectangle? If you proved some theorem or property for 'all rectangles', it would be a real waste to have to redo it all with the word 'square' added. No. The property is now automatic for squares, because squares are 'included' in the properties of rectangles, and in the word 'rectangle'.

I realize that some (many?) teachers of the early grades have not thought a lot about this choice and may have already said the opposite to their classes. However, you are right and they are wrong.

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