Defining Trapezoids

There was a recent debate on the AP Calculus EDG on the definition of a trapezoid, specifically whether a trapezoid has “exactly one pair of parallel sides” (the exclusive definition) or “at least one pair of parallel sides” (the inclusive definition).

The importance of a definition

There have been many discussions about the importance of definitions in mathematics, and in this thread, the contributors rightly noted that the inclusive definition makes parallelograms a type of trapezoid (exactly like squares are a type of rectangle), while the exclusive definition expressly forbids this because parallelograms don’t have exactly one pair of parallel sides.

Who cares, some may ask?  From one perspective, everyone is perfectly free to use either definition, but I agree with the contributor who argues that part of the discussion ought to be about what makes a definition in mathematics.  I agree completely because once definitions are accepted–especially in mathematics–they absolutely drive and constrain any and all logical conclusions one logically is able to reach.

The classification question has been discussed by Zalman Usiskin and others, but I’ll attempt a portion of my own micro-version around trapezoid classifications here.

What is a trapezoid?

First, pure trapezoids (those that can’t be called anything more specific under either definition) have precious few properties.  With nothing guaranteed about the non-parallel or congruent sides, about all you can claim is its area formula:  Area_{trapezoid}=\frac{b_1+b_2}{2}*ht where b_1 and b_2 are the parallel bases and ht is the height.  Interestingly, this same area formula also applies (with acknowledgement that b_1=b_2) to parallelograms, rhombi, rectangles, and squares, a “coincidence” ignored by proponents of the exclusive definition.

On the other hand, using an inclusive definition of trapezoids (at least one pair of parallel sides) creates the far more consistent relationship structure shown below.  (Credit due to my teaching of a UCSMP Geometry course 20 years ago.)  The more general names are toward the top of the chart and the more specific shapes are at the bottom.

So why do I care if the benefits for trapezoids are the same under either definition?  The answer lies in the properties of the shapes.

Properties

Under the inclusive quadrilateral map shown above, as soon as a more general shape is identified with a particular property, every more specific subcategory below it also has the same property.

As a first example, knowing that the area of a trapezoid is Area=\frac{b_1+b_2}{2}*ht automatically grants the same property to isosceles trapezoids and the other four quadrilaterals noted earlier.

Which quadrilaterals have perpendicular diagonals?  I know that happens first in a kite, and therefore rhombi and squares gain the property for “free”.

Once you have one set of parallel sides (a trapezoid), what else can you add to such a quadrilateral to make it more specific?  Assuming the inclusive definition, making the other pair of sides parallel, too, makes it a parallelogram where the base angles become supplementary.  If I chose instead to make the base angles congruent, the shape would be an isosceles trapezoid.  Interestingly, both additions (while seemingly independent) force the previously non-parallel sides to congruence.  But you gain something more.

In parallelograms, the diagonals bisect for the first time, a property also held only by rhombi, rectangles, and squares.  On the other hand, isosceles trapezoids are the first quadrilaterals for which diagonals are congruent, a property exclusive to isosceles trapezoids, rectangles, and squares.  Therefore, the only quadrilaterals for whom diagonals are both congruent and bisecting are rectangles and squares–the only quadrilaterals “downstream” from both parallelograms and isosceles trapezoids.  Unfortunately for the exclusive definition of trapezoids, the (isosceles) trapezoid to rectangle connection is verboten, severing a beautiful properties connection precisely at a point where it beautifully describes why rectangles are suddenly imbued with so many additional geometric relationships.

By using more inclusive definitions in mathematics, I gain far richer and interconnected ways of understanding relationships–something that certainly helps my own problem-solving and mathematical understanding.  Even better, brain research suggests that helping students see more connections between ideas reinforces understanding and recall, helping my students.

And what do I lose by eschewing the exclusive definition of trapezoids.  Nothing.

So why to others continue to adhere to to more exclusive definitions?  Because your textbooks say so or because “that’s the way I learned it” seem pretty lame to me.  Give me something practical and I might reconsider, but with nothing in sight to gain from an exclusive definition and connections, insight, understanding, and recall all enhanced by the inclusive definition, I think I’ve found the better place for my students.

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12 responses to “Defining Trapezoids

  1. David kapolka

    Chris, I had this discussion with Zalman many years ago and showed him a Sketchpad demo of two parallel lines on which I constructed a trapezoid. When I slide a vertex so as to approach a parallelogram then return to a trapezoid, he loved the dynamic visualization. This reinforced for both of us and others in the discussion why UCSMP includes the definition you and I both like as well as the hierarchy of quadrilaterals.

    David Kapolka

  2. Thanks so much for the reinforcement, David. It really is a beautiful connection away from which the exclusive definition sadly turns itself.

  3. So, in essence, it seems to me that you have redefined isosceles trapezoid from its traditional definition to make it fit appropriately with the inclusive definition of trapezoid.

    Isn’t the reason we use “isosceles” to describe isosceles trapezoid because it has the meaning that two sides are congruent? Isn’t that what we teach our students regarding triangles, the shape we first introduce them to in geometry and the shape that is the foundation for our proofs that prove so much about quadrilaterals?

    Instead, it seems that you re-define isosceles trapezoid as a trapezoid where “I chose instead to make the base angles congruent, the shape would be an isosceles trapezoid.” However, have we not used the term isogonic (or is this term less common than I thought?) with triangles to say that two angles are congruent? Why would we switch from sides to angles while staying with the same word “isosceles”?

    In other words, for triangles, we use “isosceles” to say two sides are congruent and then prove that two angles are then forced to be congruent. Yet, you seem to want to use “isosceles” for trapezoids to say that two consecutive angles are congruent (and then I’m assuming you would use that to prove there has to be a set of congruent sides). It would eliminate the base angle theorem (and its converse) for isosceles trapezoids as that now becomes its definition.

    If you truly want to use the inclusive definition of trapezoid and then exclude typical parallelograms from isosceles trapezoids (parallelograms should fit the definition of isosceles trapezoids where one set of sides is parallel while another set of sides is congruent) by focusing on the angles rather than the sides, that seems to be inconsistent with the initial use of the term “isosceles” that we introduce to our students earlier with triangles.

    Overall, it seems that by choosing an inclusive definition of trapezoid, one is then forced to use a more exclusive definition of isosceles trapezoid or a more restrictive use of the base angles theorem for isosceles trapezoids (and its converse) as well as the theorem regarding congruent diagonals for isosceles trapezoids. Thereby you do lose something in the process one way or the other.

    These are my musings. I’m willing to be convinced of the inclusive definition if someone could adequately answer those issues. Until then, I am (to use another math class’s term) in the position of failing to reject the null (definition, that is).

  4. Doug Kuhlmann

    Chris, I agree with you completely. However, the folks who would like to use the exclusive definition of trapezoid would take issue when you say you lose nothing by using the inclusive definition. One loses the ability to say “base angles of an isosceles trapezoid are congruent” and “diagonals of an isosceles trapezoid are congruent”. Parallelograms would have to considered isosceles trapezoids. For me this is not a big deal: One just needs to say “base angles of an isosceles trapezoid that is not a parallelogram are congruent.” Of course one could change the definition, not of trapezoid, but of isosceles trapezoid: An isosceles trapezoid is a trapezoid that has a pair of sides that are congruent but not parallel.

    Again, I like your diagram above.

    Doug Kuhlmann

    • I think my initial post wasn’t as clear as I intended. Once you have a “pure trapezoid” (those that can’t be called anything more specific under either definition), the next more specific shapes are either isosceles trapezoids (when the base angles are congruent) –OR– parallelograms (when the base angles are supplementary). I never intended parallelograms to be forced to be isosceles trapezoids, as indicated by the split in the diagram below trapezoids. Using inclusivity, of course there are parallelograms which are isosceles trapezoids, and vice versa.

      This makes me think of some potential student prompts.
      One student claimed a particular quadrilateral was a parallelogram. Another insisted it was an isosceles trapezoid. Could they both be right? For which quadrilaterals could both be wrong?

  5. Defining iscosceles trapezoid as one in which the base angles are congruent is a nice “hack” in my opinion… it looks pretty good to me, though.

    I think it’s a “hack” because Euclid says that isosceles triangles have two _sides_ congruent.

    Convinced me, though.

  6. Given some of the feedback above, I’m convinced my argument could be more historically consistent with a different divergence from the “pure trapezoids.” Once you have a single set of parallel sides, make the other pair of sides congruent. A little geometric investigation shows that quadrilaterals with these two properties come in two types: parallelograms (when the base angles are supplementary) & isosceles trapezoids (when the base angles are congruent). Trapezoids with congruent sides whose the base angles hold both properties are the even more specific “rectangles.”

    In the end, I still end up back at precisely the same diagram I originally posed. Because the property relationships remain the same using both approaches, and the congruent sides definition is more historically consistent, I agree that using congruent sides is a better approach. Thanks for the correction.

    The beauty here is that whether isosceles trapezoids are trapezoids with congruent base angles (guaranteeing congruent sides) –OR– are trapezoids with congruent sides (guaranteeing congruent base angles), you get exactly the same quadrilateral and properties. With this minor revision, I have absorbed the historical definition of isosceles trapezoids into the inclusive definition, potentially even strengthening it.

    I still don’t see what the exclusive argument gives.

    Here’s another nice connection, perhaps. When considering Side-Side-Angle-defined potential triangles, the side not attached to the defined angle is (in a sense) free to rotate around a circle whose radius is the length of the free side. Depending on the magnitudes of the other conditions, the “free” side could “connect” to form 0, 1, or 2 triangles (the ambiguous case because a single universal statement cannot be made about SSA information). The additional connection I see is that adding a congruent pair of sides to a pure trapezoid creates the same sort of “free” sides, in this case becoming either a parallelogram or an isosceles trapezoid, depending on the relationship between the base angles once on the free sides settle.

    THANK YOU, everyone, for your feedback.

  7. Not to beat a dead horse, but the following statement still seems to have problems: “The beauty here is that whether isosceles trapezoids are trapezoids with congruent base angles (guaranteeing congruent sides) –OR– are trapezoids with congruent sides (guaranteeing congruent base angles), you get exactly the same quadrilateral and properties.”

    Trapezoids with congruent sides doesn’t guarantee congruent base angles unless you specifically exclude non-special parallelograms from that definition. Trapezoids that have congruent sides could have those congruent sides be non-parallel (the typical isosceles trapezoid) or parallel (a parallelogram). In the first case, the base angles are congruent. In the second, the base angles are supplementary. It still seems to me that in order to use an inclusive definition of trapezoids, one must use an exclusive definition for isosceles trapezoids (or use an exclusive version of the base angles theorem and its converse and an exclusive version of the theorem that states that the diagonals of an isosceles trapezoid are congruent).

    Also, you said that “beautifully describes why rectangles are suddenly imbued with so many additional geometric relationships.” However, could we not say that the reason that rectangles are imbued with so many additional geometric relationships comes from the fact that a rectangle is defined as equiangular? Therefore, it is easily proven that a rectangle must be a parallelogram (all sets of consecutive angles are supplementary forcing parallel lines) and therefore has all the properties of a parallelogram. The only missing attribute that is now needed is congruent diagonals, which can be easily proven using congruent overlapping triangles and the SAS Congruence Theorem (a good practice for students who are still struggling with proving congruent triangles and pulling out congruent parts after proving the triangles are congruent). We have to do the same type of process to prove that the diagonals of an isosceles trapezoid are congruent, so we don’t really save any time by just imbuing the rectangle with the isosceles trapezoid’s properties rather than proving the diagonals of a rectangle are congruent in its own proof in my opinion.

    I wish that it was easy enough that both definitions (trapezoid and isosceles trapezoid) could be inclusive and that we could then still use the three theorems (base angles and its converse and congruent diagonals), but we obviously cannot. Something must give somewhere, and I think the giving comes down to definitions and personal preferences of authors and teachers.

  8. Someone (I think John Conway) favors the inclusive definition, and one of the reasons he gives is “In Calculus, have you ever heard anyone call it the Trapezoid Or Rectangle Method for integration?”

    Of course not! We all recognize that a rectangle is a special trapezoid. There’s a huge advantage there, because the area formula still works. Thus, we are all already using the inclusive definition at times, even those of us who claim to be using the exclusive definition.

    • Nice … I’d always argued this from a geometric perspective. The calculus approach is a nice additional extension.

      I think part of the reason people are so hung up on trapezoids not being parallelograms rolls back historically to Euclid. 1000s of years later, we still have problems teaching differently than we were taught.

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