Question

20 POINTS !!!!!! Determine whether side-side-angle (SSA) is a valid means for establishing triangle congruence. In this case, you know the measure of two adjacent sides and the angle opposite to one of them. If it is a valid criterion, explain why. If it is not valid, use GeoGebra to create a counterexample demonstrating that it doesn’t work and give an explanation. (Hint: Try constructing a triangle where the known angle is opposite to the shortest known side.) If you construct a counterexample, take a screenshot of your work, save it, and insert the image in the space below.

Answer 1

It's not valid because there are too many unknown quantities, meaning there is not exactly one triangle that could exist for any given SSA combination.

For a visual example, see the attached image below.

It's fairly obvious that $b\neq b'$. One is acute, the other is obtuse. In the triangle on the right, the dashed side has the same length $A$ as in the triangle on the left. This dashed side and the actual side $A$ form an isosceles triangle with congruent base angles $b'$. The angles $b$ and $b'$ are supplementary (they add to 180 degrees), or $(b')^\circ=(180-b)^\circ$.

Now by the law of sines, in the first triangle,

$\dfrac{\sin a^\circ}A=\dfrac{\sin b^\circ}B$

and that in the second triangle,

$\dfrac{\sin a^\circ}A=\dfrac{\sin(b')^\circ}B=\dfrac{\sin(180-b)^\circ}B$

but we also know that $\sin(180-x)^\circ=\sin x^\circ$. Hence $\sin b'=\sin b$.

Two different angles in this arrangement allow for two different triangles to be constructed, so having a SSA correspondence between two triangles is not enough evidence for their congruence.