Why There Is No SSA Congruence Postulate or Theorem

Date: 03/29/2005 at 07:33:19
From: Shahid
Subject: Does SSA imply congruence?

In the correspondence of two triangles, if  SSA = SSA, are the 
triangles congruent?

Is the SSA postulate applicable?  I think we use the HS postulate in
place of SSA.

Date: 03/29/2005 at 14:45:59
From: Doctor Rick
Subject: Re: Does SSA imply congruence?

Hi, Shahid.

There is no general SSA postulate (or theorem).  Unlike the familiar 
congruence theorems, further conditions must be placed on the 
triangles in order to make a valid theorem.  One such condition is the 
one you mention: if the triangles are right triangles and one of the 
congruent sides is the hypotenuse of each triangle, then SSA is valid.

Here is an example of a pair of triangles in which two sides and a 
non-included angle are congruent but the triangles are not congruent:

                                           C
                                          *
                                       /  |
                               E    /     |
                                *         |
                             /   \        |
                      D   /       \       |
                       *           \      |
                    /    \          \     |
                 /          \        \    |
              /                \      \   |
           /                      \    \  |
        /                            \  \ |
     /                                  \\|
  *---------------------------------------*
 A                                         B

Line BE is perpendicular to AE, and angles EBD and EBC are congruent. 
The triangles ABC and ABD have the same angle (CAB = DAB), and 
congruent sides (AB = AB and BC = BD), but obviously they are not
congruent.

As the statement stands, the implication ("if ... then") is not true:
the SSA = SSA condition does not imply congruence of the triangles.
Not all pairs of triangles in which SSA = SSA are congruent.

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