Deciding Which Statements a Theorem Requires

Every congruence theorem is really a checklist. SSS wants three sides matched. SAS wants two sides AND the angle trapped between them. ASA wants two angles AND the side between them. AAS wants two angles and a side not between them. HL, for right triangles only, wants the hypotenuse and one leg. If the facts you're handed don't line up with one of these exact patterns, you don't have congruence yet, no matter how close it looks.

The trap pattern is SSA: two sides and an angle that is NOT between them. That combination can be built two different ways, one triangle and a different triangle, so it never guarantees congruence. The fix is checking position, not just count. Ask where the angle sits relative to the two sides before you accept SAS. Similarly, AAA only pins down the angles, so it guarantees similarity, same shape, but says nothing about size, because you can scale a triangle up or down and keep every angle the same.

This skill connects directly to proof-writing: every congruence proof is really an argument that the given facts satisfy one of these templates, statement by statement. It also connects to similarity proofs, which relax the requirement to just AA, since angles alone fix shape. When you later use congruence or similarity to justify a construction or a coordinate transformation, this same checklist habit, naming the theorem, then confirming each required fact is actually present, is what makes the justification solid instead of just plausible.