Triangle Congruence: AAS and ASA

Date: 10/16/2003 at 13:29:34
From: Mrs. Bergeron
Subject: Triangle Congruency

My 8th grade geometry students (all 2 of them) were wondering why AAS 
(Angle, Angle, Side) is not a considered a valid method to prove two 
triangles are congruent.  

We can't find a way to create two different triangles if they have 2
angles and the non-including side congruent.  Can you help us?

Date: 10/16/2003 at 15:33:28
From: Doctor Peterson
Subject: Re: Triangle Congruency

Hi, Mrs. Bergeron.

The fact is, AAS is a perfectly good way to prove congruence of 
triangles. 

It's easy to prove: if two angles of a triangle are congruent to
corresponding angles in another, then the third angles are also 
congruent, since the sum of the angles in each has to be 180 degrees. 
So you really have all three angles congruent, and can apply the ASA 
theorem, using the two angles adjacent to the side S, in order to 
prove congruence. 

So AAS is just a quick corollary of ASA, which is probably why it is
not mentioned much.  Commonly when you need to use it you can just use
the argument above within your proof to show that all three angles are
congruent, and then use ASA; so you don't need to have AAS in your
arsenal.  But it is certainly useful to be aware of it.

Of course, SSA is not valid; you have probably seen some of our 
explanations of that fact.

If you have any further questions, feel free to write back.

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