SSS, ASA, SAS Proofs

Date: 01/21/2002 at 18:08:09
From: Sarah
Subject: Proofs

Dear Dr. Math,

Hello! I know you've answered a lot of questions about proofs, but 
I'm still having trouble with them. Can you please explain them in 
fuller detail? My problem is that I understand the ideas, but I'm 
not sure when and where to use them. I especially need help with SSS, 
ASA, SAS, etc. I get confused with what they mean and how to tell 
which one is which. 

Thank you,
Sarah

Date: 01/21/2002 at 23:37:04
From: Doctor Peterson
Subject: Re: Proofs

Hi, Sarah.

I assume you have looked through the Dr. Math FAQ on Proofs:

   http://mathforum.org/dr.math/faq/faq.proof.html   

and have read problems like these:

   Congruent Triangles in a Rectangle
   http://mathforum.org/dr.math/problems/lacey.11.11.99.html   

   Writing a Proof
   http://mathforum.org/dr.math/problems/tony.05.16.01.html   

   Triangle Proof
   http://mathforum.org/dr.math/problems/ashley.11.19.01.html   

   Congruent Parts Congruent Triangles Congruent (CPCTC)
   http://mathforum.org/dr.math/problems/brittany.11.28.01.html   

Here is a quick summary of what each method of proof means:

       SSS                SAS                ASA

        +                  +                  +
       /  \               /  \               /  \
     c/     \a          c/     \            /     \
     /        \         /A       \         /A     C \
    +-----------+      +-----------+      +-----------+
          b                  b                  b

        +                  +                  +
       /  \               /  \               /  \
     c/     \a          c/     \            /     \
     /        \         /A       \         /A     C \
    +-----------+      +-----------+      +-----------+
          b                  b                  b

    All three sides    Two sides and      Two angles and
    of one triangle    the included       the included
    agree with all     angle of one       side of one
    three sides of     triangle agree     triangle agree
    the other.         with the other.    with the other.

One way to think of it is to say that, if you were given one of these 
three sets of three numbers, it would be enough to be able to draw 
that triangle correctly. For example, If I told you angle A and sides 
b and c, you could draw line b, make a ray A degrees from it, and 
measure c units along that ray. Then you would have all three vertices 
of the triangle, and would know that you had exactly the shape I had 
in mind. If I just gave you, say, the three angles, you would have to 
choose a length for the sides, so you would not be sure. That's why 
there's no AAA congruence theorem.

The answers I recommended show how to use these in practice. If you 
find that you have the appropriate set of congruent pairs of parts 
between two triangles, then you can use that theorem to say they must 
be congruent. 

If you need more direct help, please write back and show me a sample 
problem that gave you trouble, so I can see where you get confused and 
suggest specific ideas. You can also find more examples at this 
ThinkQuest site:

   Geometry: Congruent Triangles - Math for Morons Like Us
   http://library.thinkquest.org/20991/geo/ctri.html   

- Doctors Peterson and Sarah, The Math Forum
  http://mathforum.org/dr.math/