How do you prove something is a rectangle?

Hi Sonja.

A rectangle is a quadrilateral with four right angles, so while it is useful to employ the idea of a parallelogram, it isn't necessary.

The reason is in the unstated part of both your proposals: what connects a single right angle to all angles being right?

If you show a parallelogram, you can use the properties of a parallelogram: opposite angles are equal, adjacent angles are supplementary.

If you don't show a parallelogram, you can use the property of transversals across parallel lines: corresponding angles are supplementary (which you have to iterate one extra step for the opposite angle, or rely on a quadrilateral - 3 right angles = 1 right angle).

So in my assessment, it is easier to apply the higher-order properties of parallelograms than the more fundamental transversal properties, but the difference is in this step of the proof.

An even simpler approach is this:

3) Find all four slopes and find all angles are formed by intersecting lines with slopes m and n. Show that n = -1/m,so all angles are right. Therefore this is a rectangle.

Hope this helps,
Stephen La Rocque.

PS: Check back at this URL in a couple of days; if other math consultants send other ideas, I'll post them here.