解三角问题

We know some congruence rules already: SSS, SAS, ASA, and AAS. Naturally this brings up the question: does SSA also work as a congruence rule?

Well, the answer is actually no. To show this, consider a triangle $\triangle ABC$, where we know $\angle A$, $AB$, and $BC$.

By the Law of Sines, $$\frac{BC}{\sin A}=\frac{AB}{\sin C}$$ Clearly, we now know that $\sin C$ is fixed. Thus, either $m\angle C=\theta$ or $180^\circ-\theta$ for some value of $\theta$.

The problem is that this yields two solutions for the triangle, and both are equally correct.

To confirm this issue, try using the Law of Cosines: $$BC^2=AB^2+AC^2-2AB\cdot AC\cos A$$ The problem is that this gives us a quadratic in $AC$, and a quadratic generally has two distinct roots!

Thus, we now know that SSA is not a valid congruence rule for a general triangle.

However, there are some special cases in which it does work.