Sufficiency: Which Statement Would Prove It

These questions don't ask you to write a proof. They ask you to recognize which missing piece completes one. That's a faster, more targeted skill: instead of building the whole argument, you check whether a candidate fact locks the conclusion down with no wiggle room left. The test is always the same: assume the candidate statement is true, then ask whether you could still draw a version of the figure where the conclusion is false. If you can, the statement isn't sufficient. If you genuinely can't, it is.

Congruence and similarity both run on fixed rules, so sufficiency almost always means matching the pieces you're given to one of those rules. If two triangles share an angle and have one equal side along that angle, you have angle and one side of SAS already locked in. The only fact left that finishes SAS is the length of the other side along that same angle. Once both arms of the equal angle match in length, the triangles are forced to be congruent. Any other extra fact, like an equal side somewhere else or an equal unmarked angle, leaves room for triangles that satisfy it but still come out different shapes or sizes.

The same logic shifts slightly for similarity. Side-angle-side similarity doesn't need equal lengths, it needs equal ratios: the two sides around the shared angle must scale by the same factor. So sufficiency there means solving for whatever value makes that ratio hold, not matching an exact length. The habit underneath both cases connects to a bigger idea: a figure can look a certain way without that fact being usable. Only statements written into the problem, or forced by other given facts, count as true for the reasoning.