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We're using Pearson's Geometry in my class.

As terms are defined there, Parallelograms include Rhombi (congruent sides), Rectangles (right angles), Squares (congruent sides and right angles, i.e. Rhombi that are Rectangles), and of course Parallelograms that are neither Rhombi nor Rectangles. Non-Parallelograms include Kites (consecutive sides congruent but opposite sides never congruent) and Trapezoids (exactly one pair of parallel sides), and again, Non-Parallelograms that are neither Kites nor Trapezoids.

I wonder. What's the point of defining the Non-Parallelograms that way? No Trapezoid property depends on the fact that some opposite sides are not parallel, and no Kite property depends on the fact that opposite sides are not congruent. It seems to me it would be more interesting and more natural to drop those requirements and to say that Trapezoids are quadrilaterals with at least one pair of opposite sides parallel (so that all Parallelograms are Trapezoids) and Kites are quadrilaterals in which each side is congruent to at least one consecutive side (so that all Rhombi are Kites).

What's the point in the lopsided way of defining terms, so that some categories intersect and some are disjoint?

Chaim
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  • https://www.youtube.com/watch?v=-pouOzsRLJM – guest Mar 19 '18 at 19:37
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    See https://math.stackexchange.com/a/2699714/124095, https://matheducators.stackexchange.com/a/13766/29, and https://math.stackexchange.com/a/650205/124095. – mweiss Mar 20 '18 at 04:55
  • @mweiss Thanks for linking to my question HAHAHAHA – BCLC Mar 21 '18 at 12:04
  • @mweiss What about this Why do we have circles for ellipses, squares for rectangles but nothing for triangles? ? – BCLC Mar 21 '18 at 12:05
  • @BCLC I agree that all of these linked questions are related, but I think that this question about circles for ellipses etc. is partly about the fact that certain mathematical ideas arose in the less rigorous, less systematic thinking of ordinary people, and mathematicians later arrived on the scene and tried to make the best use of the materials to hand. Occasionally I have a student suggest that if x=4, then 2x=24, as if the algebra were a cryptogram. I suppose that such distinctions reflect the evolution of notation used first for arithmetic and later for algebra. – Chaim Mar 22 '18 at 11:55
  • @BCLC Apparently lots of people reach age 15 thinking that quadrilaterals are either rectangles or squares. So their system doesn't start with the bifurcation of right angles vs. non-right angles; it starts and ends with congruent or non-congruent sides. Their trouble in pulling back to broader categories, to includes squares as rectangles & exclude non-rectangular quads, is a psychological problem. But I intended to ask a more logical question. Given the fact that they're only finding out from me about kites, parallelograms, rhombi and trapezoids, what would be the most elegant way to go? – Chaim Mar 22 '18 at 12:03
  • @mweiss Please see my previous two comments. – Chaim Mar 22 '18 at 12:04

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