Proving a quadrilateral is a rectangle

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Quandaries & Queries

Question from Sonja, a teacher:

I was having this discussion with another teacher and we need a third opinion. When you are trying to prove a quadrilateral is a rectangle which method should you use:
1) Prove the shape is a parallelogram by doing slope 4 times by stating that parallel lines have equal slopes. Then proving a right angle by stating that perpendicular lines have negative reciprocal slopes.
2) Doing the slope 4 times and stating that the shape is a rectangle because opposite sides are parallel because of equal slopes and it contains a right angle because og negative reciprocal slopes.

I guess the real question is do you have to first state that the shape is a parallelogram?

Thanks

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.

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