### Quadrilateral relationships - October 1995

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Take seven specific quadrilaterals and draw a diagram that shows the relationships between them. The quadrilaterals are:

1. kite
2. parallelogram
3. rhombus
4. trapezoid
6. rectangle
7. square

To make this more interesting, use the following definition of a trapezoid: "A trapezoid is a quadrilateral with at least one pair of opposite sides parallel." (You might be familiar with the definition that says it can only have one pair of parallel sides.)

Let's look at an example of what I mean using triangles. If we take an acute triangle, an equilateral triangle, an isosceles triangle, and a scalene triangle, the diagram might look like this:

```                        triangle
/     \
/       \
acute       isosceles
\       /
\     /
equilateral
```
So any theorems that apply to an isosceles triangle - "base angles are congruent", etc. - are also true of an equilateral triangle. An equilateral triangle is certainly acute, and both isosceles and acute triangles are surely subsets of triangles, but an acute triangle doesn't have to be isosceles, and an isosceles triangle isn't always acute.

So do a similar thing with the quadrilaterals. It will help to write out definitions for everything first. Good luck!

Annie says:

Picking a winner this month was a bit more of a challenge than it was in September! 28 teams submitted solutions, and 18 of them had at least the diagram correct.

Here's how I narrowed it down: I was looking for three things - the correct diagram, definitions of the shapes (which might affect the diagram), and an explanation of why the shapes are related in the manner indicated in the diagram. I felt that only three of the submissions hit on all three parts, and they are the ones that have been chosen as either winners or honorable mention. Remember when you're solving a Project of the Month: while I care about the answer itself, I care more about how you got it and why you think it's right.

I have to say that I was generally impressed by the solutions. I thought it was a hard problem so early in the year, but that with a little patience and good research skills, students might be able to pull it off. My confidence has been rewarded!

A couple of words about my posing of the problem. While I provided a definition of "trapezoid" that I wanted folks to use because it makes things more interesting, I should have provided a similar note for the kite. The three solutions here use two different definitions of kite. Both say that a kite has two pairs of congruent sides, but one goes on to say that opposite sides are not congruent. You can read the solutions to see how that affects the kite in the hierarchy.

I didn't really mean to include scalene quadrilateral as a whole separate thing. I really meant "general" quadrilateral, but it worked out okay - the scalene really is a branch of the hierarchy to itself, which is how the students treated it.

I have provided specific comments after each of the solutions, but I did want to make a general comment about something that was left out, but which I alluded to in the problem. Part of the order imposed by a hierarchy implies that theorems that are true of a higher item will be true of those items underneath it. For example, I gave the isosceles triangle theorem that "base angles are equal" as something that certainly works for equilateral triangles as well. No one looked at this as a means of classifying the quadrilaterals. They went purely on definitions (or characteristics?).

Here's an example of what would have made me very happy: "In a parallelogram the diagonals bisect each other. This is consequently true for the rhombus, rectangle, and square. In a kite, the diagonals are perpendicular, which is true in the rhombus and square, but not the rectangle or parallelogram, so the rhombus and square must lie under the kite, but the rectangle and the parallelogram can't."

There was an interesting discussion about this very thing on the Geometry Forum a while back - you can find it by searching our site for "quadrilaterals." Several people found that discussion, I think, because in their solutions the diagrams included the isosceles trapezoid, as it did in the discussion, but which I didn't include in the problem. However, those students failed to provide any explanation for their diagram, so they lost out :-)

We have two winners and one honorable mention. Each of the winners will receive a certificate and a highly coveted Geometry Forum t-shirt. The honorable mention will receive a certificate and an equally coveted Geometry Form key chain. The honorable mention falls short only in that it didn't give as many reasons why the hierarchy is the way it is.

Congratulations to the winners! The Project of the Month is shaping up really well this year.

The solutions, as well as my comments on each, follow.

### October Project of the Month

#### WINNERS: Marty Klein, Jacqulyn Law, Asa Sharma, and Jesse Chui Grade 9, College Park High School, Pleasant Hill, California

```From: "Judy M. Young"

Marty Klein
Jacqulyne Law
Asa Sharma
Jesse Chui

College Park High School, Pleasant Hill, California - Grade 9

QUADRILATERAL should come first because it is the overall term
for all of these variations.  All of the figures below have four
sides.

There should be three branches stemming from QUADRILATERAL.
They are: SCALENE QUADRILATERAL, TRAPEZOID, and then KITE.
specific enough to limit what you draw.  A SCALENE QUADRILATERAL
cannot have any equal sides.  This branch does not continue,
because any other form of a Quadrilateral (i.e., SQUARE, KITE) has
at least two equal sides, or at least some sort of restriction on
being parallel.  Because of this, nothing can follow directly under

We arrived at the other two branches (TRAPEZOID and KITE) because
all of the rest of the Quadrilaterals fall under two categories:
those pertaining to being parallel, or those having to do with
length or equalness.  The TRAPEZOID is the least restrictive in
the parallel "group" - it only needs one pair of opposite sides
parallel.  Following TRAPEZOID should be the PARALLELOGRAM.  This
calls for two parallel sides.

Let's stop for a moment and go back to the beginning.  The other
branch off of QUADRILATERAL is the KITE.  Although the criterion
is strict, it is still less restrictive than the remaining
quadrilaterals in the length and equalness "group."  KITES call
for two consecutive sides of equal length.

The next quadrilateral after KITE has to be a RHOMBUS--this one
needs all four sides to be equal.

You can now go back and draw a line from PARALLELOGRAM to RHOMBUS.
When you think about it, the relationships between these two
polygons are very similar.  PARALLELOGRAMS need two pairs of
parallel lines, and RHOMBUS'S must have four equal sides.  These
two kind of go hand-in-hand, but PARALLELOGRAMS just can't boast a
specific enough definition to earn a spot any lower on our
hierarchy.

You can now draw a line from PARALLELOGRAM down to our next
quadrilateral, the RECTANGLE.  A RECTANGLE has four right angles,
which implies that there are two pairs of parallel sides.

Now write SQUARE below and in between RHOMBUS and RECTANGLE.
Squares not only have four equal sides (not unlike the RHOMBUS),
but they also have four right angles and two pairs of parallel
sides (like the RECTANGLE), so you must now connect SQUARE to
both.

* CHART *

The finished hierarchy should look like this:

/            |           \
/             |            \
|               |
|               |
|       PARALLELOGRAM
\           /  \
\         /    \
RHOMBUS    RECTANGLE
\          /
\        /
SQUARE

***********************************************************************
* Judith M. Young                          College Park High School   *
* Mathematics/Senior Experience Exchange   201 Viking Drive           *
* 510-682-7670 X3233                       Pleasant Hill, CA 94523    *
* E-mail  jyoung@cello.gina.calstate.edu   FAX: 510-676-7892          *
***********************************************************************
```
COMMENTS: Any solution that is not only correct but that is written in complete sentences scores points with me. I like the way they structured the answer, working the definitions in with the explanation, though I might have put the diagram first so that we would know where they're going, as well as including a statement of the problem at the very beginning (which also helps prevent having the diagram come across over a page break!). Having the names of the quadrilaterals in CAPS is a good idea. In several cases they might have explained a bit more clearly why something was under something else - the parallelogram and rhombus relationship, for example. The best part, and one which should serve as a model for the rest of the explanation, is the last paragraph - you just can't argue with reasoning like that.

#### WINNER: Carolyn DiMaria Grade 9, Mount St. Joseph Academy, Flourtown, Pennsylvania

```From: ruth@mathforum.org (Ruth Carver)

Carolyn DiMaria - Mt. St. Joseph Academy - freshman

[Carolyn sent her solution as a Sketchpad file, but I took out the
text and reproduced the diagram.]

The problem is: draw a diagram that shows the relationships
between: a kite, parallelogram, rhombus, trapezoid, scalene
equilateral, rectangle, and a square.  Explain how you got those
relationships.

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

Now  I have to try and explain how I got this relationship map.

First, I looked up the Mathematical definitions of each of the seven
figures. That really helped a lot because most of the definitions
said things like parallelogram with.... or quadrilateral with...
So from there, I just filled in what I could. I've included the
definitions, so you can see how I got these.

KITE - Quadrilateral that has two pairs of congruent sides, but
opposite sides are not congruent.
PARALLELOGRAM - quadrilateral with opposite sides equal and
parallel.
RHOMBUS - equilateral parallelogram usually having oblique (not
right) angles.
TRAPEZOID - quadrilateral having at least two sides parallel.
SCALENE QUADRILATERAL - having three sides of unequal length.
RECTANGLE - parallelogram with all right angles.
SQUARE - rectangle with all four sides equal.

Okay, now for my professional sounding explanation.

None of these seven figures has any properties of a kite, so that
stands alone.  The trapezoid can be a scalene quadrilateral, but
the line is not drawn because the others under the trapezoid are
not related to a scalene quadrilateral. All objects with the
exception of the kite and scalene quadrilateral can be a
trapezoid. The rhombus can be a parallelogram because the opposite
sides are equal and parallel with oblique angles.  A rectangle has
opposite sides equal and parallel, with all angles right.  And a
square is both a parallelogram and a rectangle and also a rhombus.
Its opposite sides are equal and parallel, and its angles are
right.   Just by looking at the definitions you can deduce this
relationship tree.

And if you can't understand what I'm trying to say, I apologize.
HAPPY MATH ! ! ! !
```
COMMENTS: First, note that Carolyn's kite isn't attached to her rhombus. Based on her definitions, that's exactly right. The first group's is attached because they used a less restrictive definition (it's the difference between "has" and "has at least"). I like that the problem is stated right at the beginning and the diagram is shown. The paragraph right after the diagram saying how looking up the definitions helped a lot is really nice - it gives you an idea of how she tackled the problem. And her explanations really emphasized why things were alike so that we'd see the links more clearly, though I'd like it if she gave some reasons why things can't be related to other things.

#### HONORABLE MENTION: Susan Tull Grade 9, Mount St. Joseph Academy, Flourtown, Pennsylvania

```From: ruth@mathforum.org (Ruth Carver)

Susan Tull

x               x            x
x
Parallelogram
x             x
Rectangle          Rhombus
x                  x
x           x
x     x
x
Square

Trapezoid - has at least one pair of opposite sides parallel
Parallelogram - both pairs of opposite sides parallel
Scalene Quadrilateral - no congruent sides
Square - 4 right angles; 4 congruent sides
Rectangle - 4 right angles
Rhombus - 4 congruent sides
Kite - 2 pairs of congruent sides, but opposite sides are not
congruent

A quadrilateral can be any of the shapes listed.  A square is
always a rhombus, a rectangle, a parallelogram, a trapezoid, or a
quadrilateral.  A rectangle or a rhombus can always be a
parallelogram, a trapezoid, and a quadrilateral.  A parallelogram
will always be a trapezoid and a quadrilateral.  A scalene
quadrilateral can be a trapezoid but I didn't know how to show it
in the diagram.  A kite can only be a quadrilateral as with the
scalene quadrilateral.  This is why I drew my diagram as displayed
above.
```
COMMENTS: Susan explains which things can be which, but doesn't really explain why quite as clearly, and doesn't mention what can't be related to what - what properties of those quadrilaterals should we concentrate on?

- Annie