In congruency tests, must the matching angle be the one in between the two sides? When proving congruency, one of the classic tests is SAS, where the angle is between the two matching sides. Usually, it is taught that the angle must be between the two sides for this to work. 
Is it really true that the angle absolutely must be between the two sides to work?
For example, if I have the following triangle (drawn to scale, so the double dashed line is longer than the single dashed line), then what other kind of triangle can be drawn that is not congruent to this, yet has two matching sides and this angle?

I understand how the counter example can be made if the double dashed line is longer than the single dashed line (as in the diagram below), but not for the above case.

 A: You are right, let the two bottom angles from left to right $A,B$, and the two sides opposite to them $a,b$.
Then $A+B<180^{\circ}$, $a\leq b\implies B\leq A \implies 2B\leq 180^{\circ}$. 
Since $sin(B)=sin(A)\cdot{b\over a}$ is fixed, $B\leq 90^{\circ}$ is sufficient to show $B$ has only one value. The congruence then comes from $AAS$.
However if $a>b$ then $B$ can take its value $sin^{-1}(sin(A)\cdot{b\over a})$ but it can also take $180^{\circ}-sin^{-1}(sin(A)\cdot{b\over a})$ as well.
A: The second diagram is a counter example to the claim that SSA would lead to congruence. Although congruent triangles do have SSA in common, that does not mean that every two triangles that have the same SSA must be congruent. 
On the other hand, any two triangles with SAS in common will definitely be congruent. There can't be any counter example, like your second diagram, that would prove this otherwise. 
Perhaps Angle-ShortSide-LongSide necessarily leads to congruence, but not Angle-Side-Side in general.
A: I guess what you want to say is ASS cannot be used to deduce a congruency in general. However, if we can further show that (for both the triangles in question) the side opposing the given angle is larger than the other given side, then congruency is still true.
I think you are right and you can add that as a special case of ASS.
