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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

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

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

+
. \\
.  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 -

>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

+--------+          +
/         |         /     \
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
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

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/

Associated Topics:
High School Geometry
High School Triangles and Other Polygons

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