Student: So, what's the difference between a rectangle and a square?

Mentor: Well, let's see. They both have 4 sides, but are the sides the same length?

Student: Um, in a square it looks like all the sides are the same. But in a rectangle, only two of the sides are the same. I mean, the top and the bottom are the same and then the sides are the same, but the top and the side are different in a rectangle.

Mentor: That's good. So all the sides of a square are congruent, or have equal lengths, and only the opposite sides of the rectangle are congruent.

Are there any other parts of the squares and rectangles that seem to be the same?

Student: The opposite sides are parallel, like in a parallelogram, only they go straight up in the air. They don't lean to the side like most parallelograms do.

Mentor: Good, rectangles and squares ARE parallelograms. You noticed, though, that the angles in the rectangle and square are the same. Do you remember what we call angles that are straight up like that?

Student: I think that they are called right angles?

Mentor: Right! (ha ha) That is to say, you are correct. So rectangles and squares have how many right angles?

Student: They have 4 right angles.

Mentor: Good. Now, can we make a general rule to determine what defines a rectangle and a square?

Student: First, each shape has to have 4 right angles.

Mentor: Good. What do we need to say about the sides?

Student: We need to say that they are parallel and that in a square all four sides are congruent while in a rectangle only the opposite sides are congruent.

Mentor: Okay, so a rectangle must have opposite sides that are parallel and equal and four right angles. A square must have opposite sides parallel, all sides congruent, and four right angles.

We could also say that a rectangle is a parallelogram with four right angles, while a square is a parallelogram with four congruent sides and four right angles.

But wait! Does that mean that a square is a rectangle?

Student: Well, by our definition it is since it has opposite sides that are congruent and parallel and it has four right angles. Did we define it wrong?

Mentor: No, actually you did it perfectly. A square IS a rectangle. Can anyone tell me why a rectangle is NOT a square?

Student: A rectangle is not a square because it does not have four congruent sides.

Mentor: Very good!

Now we can talk about trapezoids!

Please direct questions and comments about this project to Addison-Wesley math@aw.com
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