Wednesday, January 29, 2014

12: Trapezoids ...

from Kevin Shonk (via email)

What makes for a better definition?
  • A trapezoid has at least 1 pair of parallel sides 
    • OR 
  • a trapezoid has exactly one pair of parallel sides? 
 Categorize it:
  • Is a parallelogram a type of trapezoid or is a trapezoid a type of parallelogram?


  1. I like how the question is phrased in that you don't ask what is the "correct" definition, but which makes for a better definition. Personally, I tend to be a fan of the more inclusive definition. So equilateral triangles are also isosceles and squares are also rectangles (which are also parallelograms which are also trapezoids). This way if I prove a statement about trapezoids I know it will also hold for other parallelograms as well. Off the top of my head I can't think of any useful statements that would be true for quadrilaterals with exactly one pair of parallel sides, but not true for parallelograms.

  2. There are a few characteristics of a trapezoid. More for an isosceles trapezoid. And if parallelograms were a sub-category of trapezoids, they would actually be a sub-category of isosceles trapezoids. But then you could not talk about "congruent base angles" and lines of symmetry of an isosceles trapezoid since parallelograms don't have that. But isosceles trapezoids and parallelograms do share similar traits regarding diagonals if I remember correctly. And you can also compare the area formula for trapezoid to area formula for parallelogram. Good conversations!

    A few more conversations about quadrilaterals:
    *is a rhombus a sub-category of a kite?
    *different countries use different definition of "trapezoid" some also use a word "trapezium". I always forget which one is the same as "isosceles trapezoid".

  3. Consider the law of continuity. Lets say we have a right trapezoid such that the non parallel/perpendicular side is protruding out in the top right corner. As we move that corner in toward the left, the shape is still a trapezoid at every point except possibly the point where it would make the side parallel to its opposite side. In my opinion it does not make sense to have this random gap in determining a trapezoid. As you move that corner there is no logical explanation to say, its a trapezoid, its a trapezoid, now it's not (because its a parallelogram), but ohh as you keep going it becomes a trapezoid again. Therefore the definition should be "at least 1" rather than "exactly 1"

  4. Beautiful question! I prefer to think that parallelograms are a subclass of trapezoid in which both pairs of sides are parallel (i.e. opposite angles are congruent) and that isosceles trapezoids are a subclass of trapezoids in which both pairs of base angles are congruent. Then isosceles trapezoids can take on all of their special properties (particularly congruent diagonals), and the intersection of the set of all isosceles trapezoids and the set of all parallelograms is the set of all rectangles.

  5. After some consideration I have to say I like definition 1 for a trapezoid. Many things in mathematics are connected to one another and I feel this inclusive definition of a trapezoid would speak to that. Much in the same way (as someone stated above) that a square is considered a rectangle, in choosing this definition we can consider squares and other parallelograms to be a type of trapezoid (although this may bother some folks!)