## 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!