Why can't there be an axiom of congruency of triangles as A.S.S. similar to R.H.S.?

Question

I think that, if we consider 2 triangles, whose one angle, the side opposite to that angle and any other side are equal, then the two triangles are congruent. If I am wrong, can someone please construct 2 non-congruent triangles with these conditions?

James Atkins · Accepted Answer

<p><img alt="enter image source here" src="https://tutor.hix.ai <br>
(details below)</p> If #C# is the center of a circle, the #abs(CB)=abs(CD)# By construction
#color(white)("XXX")/_BAC=/_DAC# In triangles #triangle BAC# and #triangle DAC#
#color(white)("XXX")/_BAC=/_DAC#
#color(white)("XXX")abs(AC)=abs(AC)#
and
#color(white)("XXX")abs(CB)=abs(CD)# So we have an A.S.S. arrangement
but
#color(white)("XXX")triangle ACB# is not congruent to #triangle ACD#

Autumn Armstrong · Answer

undefined The A.S.S. (Angle-Side-Side) condition is not a valid criterion for triangle congruence because it can lead to ambiguity and does not uniquely determine a triangle. Unlike the R.H.S. (Right Angle-Hypotenuse-Side) condition, which is a valid criterion for congruence in right triangles, the A.S.S. condition may result in multiple triangles with different shapes and sizes that satisfy the given conditions.

Specifically, the A.S.S. condition does not account for the fact that given two angles and a side length, there may be more than one triangle that can be constructed. This ambiguity violates the requirement of a valid axiom, which should lead to a unique geometric outcome.

Therefore, the A.S.S. condition cannot be used as an axiom of congruency for triangles because it does not provide sufficient information to uniquely determine congruent triangles.