Why does the ASS (or SSA) can't be used to determine triangle congruence?
Why is not the ALL or LLA a case of congruence between triangles?
This question can raise a lot of doubt. Thus, we will explore the activity in a pratical way, so that the explanation is clearer.
Opening Remarks
In the worksheet of "Cases of Congruence" we can see that there are four cases: SAS (side-angle-side), ASA (angle-side-angle), SSS (Side-Side-Side) and AAS (angle-angle-side). It was natural to think that ASS (angle-side-side) or SSA (side-side-angle) would be possible. Why can't we use these situations as congruence cases? In other words: Why is that two triangles, that have a congruent angle, an congruent adjacent side and an opposite congruent side, not necessarily be considered congruent?
SOLUTION HYPOTHESIS
As the exercise already states, ALL cannot be considered a case of congruence, so we must prove that this is true. In order to prove this, we can simply show a case in which two triangles have a congruent angle, an congruent adjacent side and an opposite congruent side, and are not congruent.
Construction of a congruent triangle to ΔABC in which the only measures given are of two sides and of an angle.
Analysis
Move points A, B or C and see if the triangles remain congruent. The question is, Would it be possible to build another triangle whose sides are congruent to AC and CB and that also have a congruent angle. Note that this new triangle is not congruent to ΔABC.
Construção de dois triângulos que possuem ordenadamente congruentes um ângulo, o lado adjacente e o lado oposto e não são congruentes.
Analysis
Would it be possible to find another triangle that had an angle, an adjacent side and an opposite side, all of them congruent, and that was not congruent to the triangle ABC?