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### 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

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

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

|                                                     |
|         +--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/
```
Associated Topics:
High School Discrete Mathematics
High School Logic
High School Sets
High School Triangles and Other Polygons
Middle School Logic
Middle School Triangles and Other Polygons

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