Why does SSA not work? (select all that apply if more than one is true) because if you have two congruent sides and the angle not included, they will always be congruent triangles you can have two different triangles where the two side are congruent and an angle NOT included are congruent, so we can't say that ALL trianlges will be congruent by SSA because it does not hold true ALL all the time

Answer 1

The reason the Side-Side-Angle (SSA) condition does not guarantee triangle congruence is due to the ambiguous case. Here are the key reasons:

1. Ambiguity of SSA: Given two sides and a non-included angle (SSA), you can have situations where two different triangles can be formed. This is known as the "Ambiguous Case" of the Law of Sines, where it's possible to have two triangles, one triangle, or no triangle at all depending on the particular values of the sides and angle involved.

2. Non-unique Solutions: Because the angle is not included between the two sides, the configuration can yield multiple solutions. For example, if you have two sides of length \( a \) and \( b \), and an angle \( A \) that is not between those two sides, you can end up with two different triangles that satisfy the SSA condition.

These points highlight why the SSA condition cannot be used to conclude that triangles are congruent in all cases. Thus, it is not a valid method for establishing triangle congruence.