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# Triangles and Trapezoids

Debating definitions has long been one of the favourite pastimes of math teacher Twitter. (see, for example, #sandwichchat or #vehiclechat). Recently, and in a move of pedagogical brilliance, the collegial tone of such debates was soured by an ongoing feud between Shelby Strong and Zak Champagne.

The object under debate: The trapezoid.

Both teams made their case and canvassed for support. Shelby argued for an inclusive definition, Zak argued for an exclusive one, and math teachers aligned themselves in one camp or the other: #TeamInclusive or #TeamExclusive. (You can pledge your allegiance in apparel form here or here.)

I was more than happy to take my place on the sidelines, just hoping both teams had fun, until …

Full disclosure: This post does not pick a team. Before I make such a character-defining decision, I’d like to have all the information. Therefore, this post asks a critical question to both #TeamInclusive and #TeamExclusive:

Where do you stand on triangles?

If we restrict consideration to triangles and quadrilaterals (due to their familiarity in the grade school outcomes and standards), it quickly becomes apparent that the scalene triangle is the only shape in this set that is defined based on what is does not have, while all others are defined based on what they do have. The definer, by adopting these definitions, is already implying the poor scalene triangle is wanting.

Pay attention to how both trapezoid teams define and classify trapezoids based on attributes they boast. #TeamInclusive claims that trapezoids have at least one pair of parallel sides, and #TeamExclusive claims that trapezoids have exactly one pair of parallel sides. Contrast these definitions to the widespread definition of the the scalene triangle, wallowing alone in its deficiencies. In case you are not familiar, a scalene triangle is conventionally defined by what it doesn’t have: any congruent sides.

• A scalene triangle is a three-sided polygon with no congruent sides.

So big deal? Maybe this is just a special take on an exclusive definition? Perhaps such a claim would abate my concern if, conventionally, the same people claiming to be #TeamExclusive with scalene triangles weren’t so #TeamInclusive with equilateral and isosceles triangles! That’s right; many a geometric two-timer holds an inclusive definition for isosceles and equilateral triangles all the while maintaining an exclusive one for the scalene triangle alone.

• An isosceles triangle is a three-sided polygon with two congruent sides.
• An equilateral triangle is a three-sided polygon with three congruent sides.

Sound familiar? An equilateral triangle is defined as having three congruent sides, which–for some–satisfies the isosceles requirement of having two congruent sides. After all, for #TeamInclusive, having three congruent is having two congruent. However, the scalene triangle is blatantly excluded from the hierarchy because it is defined exclusively as having no congruent sides.

This just might be the way things are. It’s a big, bad 2D plane out there, and there will be casualties in any system. However, that doesn’t stop a guy from demanding consistency. As a proverbial olive branch, I’ve taken the liberty of offering two, new sets of definitions for triangles–one consistently #TeamInclusive and the other consistently #TeamExclusive.

##### #TeamInclusive
• A scalene triangle is a three-sided polygon with at least one side of equal length.
• An isosceles triangle is a three-sided polygon with at least two sides of equal length.
• An equilateral triangle is a three-sided polygon with at least three sides of equal length.
##### #TeamExclusive
• A scalene triangle is a three-sided polygon with exactly zero pairs of sides of equal length.
• An isosceles triangle is a three-sided polygon with exactly one pair of sides of equal length.
• An equilateral triangle is a three-sided polygon with exactly three pairs of sides of equal length.

The choice is now yours. Ultimately, until the world addresses its misgivings regarding the scalene triangle, it appears that my allegiance belongs to neither team. I’m not #TeamInclusive or #TeamExclusive. I’m #TeamScalene.

NatBanting