Venn Diagram to Classify Quadrilaterals

Date: 01/02/2003 at 14:29:28
From: John
Subject: Venn Diagram to classify Quadrilaterals

I am looking for a Venn diagram that will accurately display the 
relation among trapezoids, parallelograms, kites, rhombi, rectangles, 
and squares.

Is a square also a kite?

The textbook I am using defines a kite as a quadrilateral having at 
least two pairs of adjacent sides congruent, no sides used twice in 
the pairs. Why the "at least two pairs" and the "no sides used twice"? 

Date: 01/04/2003 at 23:26:44
From: Doctor Peterson
Subject: Re: Venn Diagram to classify Quadrillaterals

Hi, John.

Here's a tree diagram, from which you can make a Venn diagram:

   Definition of a Trapezoid
   http://mathforum.org/library/drmath/view/54901.html 

                      Quadrilateral
                      /           \
                    /               \
              Kite                    Trapezoid
               |                      /       \
               |                    /          \
               |              Parallelogram   Isosceles
               |              /       \       Trapezoid
               |            /           \       /
               \          /               \   /
                 Rhombus                  Rectangle
                      \                  /
                        \             /
                          \        /
                           Square

So let's see if this turns into a nice Venn diagram:

    +--quadrilateral--------------------------------------+
    |                                                     |
    |         +--trapezoid------------------------------+ |
    |         |                                         | |
    |         +--parallelogram-----------+              | |
    |         |                          |              | |
    |         |           +--------------+--iso trap--+ | |
    |         |           |              |            | | |
    | +-kite--+-----------+------+       |            | | |
    | |       |       rhombus    |       |            | | |
    | |       |           |      |       |            | | |
    | |       |           |square|       |            | | |
    | |       |           |      |       |            | | |
    | |       |           |   rectangle  |            | | |
    | |       |           +------+-------+------------+ | |
    | |       |                  |       |              | |
    | +-------+------------------+       |              | |
    |         |                          |              | |
    |         +--------------------------+              | |
    |         |                                         | |
    |         +-----------------------------------------+ |
    |                                                     |
    +-----------------------------------------------------+

A label on an edge refers to everything inside; a label inside a 
region refers to that region only; a label straddling two regions 
names everything in those regions. So some quadrilaterals are kites, 
some are trapezoids, and some are neither; some trapezoids are 
parallelograms, some are isosceles, and some are neither; 
parallelograms that are also isosceles trapezoids are rectangles; 
parallelograms that are also kites are rhombuses; those that are 
both are squares.

Note that I am using the inclusive definition of a trapezoid, which 
not everyone might agree with: a parallelogram is a kind of trapezoid, 
in which not only one, but two pairs of sides are parallel.

Your book's definition of a kite seems awkward. They want to make sure 
you don't count three consecutive congruent sides as two pairs, so 
they say you can't use the same side twice; and I can't imagine why 
they bother saying "at least two pairs," since once you've chosen two 
disjoint pairs you've used up all the sides. Maybe they want to 
explicitly allow for the square, in which there are four pairs of 
congruent sides, giving two ways to choose two that are disjoint to 
fit the other rule. In any case, the square is a kite.

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/