Theorems for Quadrilaterals

Date: 11/12/1999 at 01:04:10
From: Bethmarie
Subject: Using the SAS, ASA, SAS, & SSS methods

Dear Dr. Math,

I am currently in an honors geometry class and have a project due. I 
keep reading it over and over but I don't know how to solve it. Can 
you please help me? Here's my project:

Use the following exercises to investigate methods of proving 
congruence of quadrilaterals similar to the ASA, SAS, SSS congruence 
postulates for triangles.

1. Does the SAS method of proving congruence work for quadrilaterals? 
That is, if 2 quadrilaterals ABCD and EFGH have AB congruent to EF, 
angle ABC congruent to angle EFG, and BC congruent to FG, must the two 
quadrilaterals be congruent? If so, give a proof. If not, draw a 
diagram that shows the method does not work.

2. For each of the following possible methods of proving congruence of 
quadrilaterals, either prove that the method works, or draw a picture 
that shows it does not imply congruence. As with triangles, the parts 
represented by the letters in each abbreviation are consecutive as you 
move in one direction around the quadrilateral.

     A) SASA     B) SASAS     C) SSSS     D) SASSS

3. Find at least two more methods of proving congruence of 
quadrilaterals. Prove that your methods work.

Can you give me some ideas on how to prove these congruencies?

Thank you.
-Bethmarie

Date: 11/12/1999 at 13:02:08
From: Doctor Peterson
Subject: Re: Using the SAS, ASA, SAS, & SSS methods

Hi, Bethmarie.

>Use the following exercises to investigate methods of proving 
>congruence of quadrilaterals similar to the ASA, SAS, SSS congruence 
>postulates for triangles.

This sounds like an interesting exercise, but it does take some work, 
some imagination, and a familiarity with congruence. Let me start by 
reviewing triangle congruence informally, so you can see what lies 
behind it:

The three congruence theorems (or postulates, in your text's approach) 
are three ways of choosing three corresponding parts of a triangle, 
which are sufficient to construct the triangle. Here they are:

         +                +                +
        /  \             /A \             /  \
      S/     \S        S/     \S         /     \
      /        \       /        \       /A      A\
     +-----------+    +-----------+    +-----------+
           S                                 S

The first, SSS, says that if I know the lengths of three sides, 
there's only one kind of triangle I can make from them; any such 
triangle is congruent to any other. Imagine trying to fit three sticks 
together to make a triangle and you can see what's going on.

The second, SAS, says if we have two lengths and the angle between 
them, we've determined the triangle completely. Imagine taking two 
sticks and putting them at a certain angle to one another; there's 
only one way to place the other side to make the triangle.

The third, ASA, says if we know a side and the angles the other two 
sides make with it, we know what the triangle is. Imagine taking one 
stick, and drawing a line at a specified angle to it on the left, and 
another line at a specified angle on the right; where these lines 
cross determines where the other vertex has to be.

There are only three other ways you could choose three parts of a 
triangle:

         +                +                +
        /  \             /A \             /A \
      S/     \S         /     \          /     \
      /       A\       /A       \       /A      A\
     +-----------+    +-----------+    +-----------+
                           S

Of these, the middle one, AAS, works, because you can figure out what 
the missing angle is (how?) and use it to make an ASA. The first, SSA, 
almost works; but if you tried to make a triangle knowing these three 
things about it, you might make this triangle instead:

         +
        . \\
       .  S\ \S
      .     \ A\
     ........+---+

And if you know only AAA, you know the shape, but not the size; this 
has the same angles, but is only similar, not congruent:

       +
      /  \
     +-----+

What you want to do is to repeat this sort of thinking - what 
information is sufficient to construct a triangle, and what isn't - 
in the case of quadrilaterals.

>1. Does the SAS method of proving congruence work for quadrilaterals? 

Just imagine you have two sticks and an angle, as above, and see if 
that determines the quadrilateral:

         +--------+          +
        /         |         /     \
      S/          |       S/            +
      /A          |       /A           /
     +------------+      +------------*
            S                   S

It's pretty obvious that, since you know nothing about the other two 
sides, the fourth vertex can go anywhere - you can easily have two 
non-congruent quads that have SAS the same. Just draw one example like 
this to show it doesn't work. A counterexample is all you need for a 
negative proof like this.

>2. For each of the following possible methods of proving congruence 
>of quadrilaterals, either prove that the method works, or draw a 
>picture that shows it does not imply congruence.
>
>     A) SASA     B) SASAS     C) SSSS     D) SASSS

Here you have at least four pieces of information. Try constructing a 
quad in each case and see if you're forced to build it a certain way 
or have several choices. For example, take (C). If I have four sticks 
and attach them at their ends, they'll flop around all over the place 
- if I change an angle, they'll move freely but still be a 
quadrilateral. So SSSS is NOT enough to prove congruence. See if any 
of the others work. Then you'll have to prove them; you'll do that 
using triangles - start with some corner of the quad that has three 
parts given, and draw the diagonal to form a triangle. You should be 
able to prove that this triangle is determined by what you're given, 
and then move on to show the other half of the quad is also known.

>3. Find at least 2 more methods of proving congruence of 
>quadrilaterals. Prove that your methods work.

If you run out of S's and A's to use, try D's - diagonals. Those can 
be very useful to make a quad rigid.

When you get some good ideas, if you still need help with the proofs, 
feel free to write back.

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