Proving Congruence: SSS & SAS

You do not need all six pairs to prove two triangles congruent. Three of the right pieces are enough.

SSS: all three pairs of sides congruent. Triangles are rigid, so the three side lengths lock the shape completely.

SAS: two pairs of sides and the pair of angles *between* them. The word "included" is doing real work — the angle must sit between the two sides you used.

SSA is not a criterion. Two sides and a non-included angle can produce two genuinely different triangles, which is why it fails. (The one exception is a right angle, which gets its own rule, HL.)

In proofs, look for a shared side. It is congruent to itself by the Reflexive Property, and it is very often the third piece you need.

SSS and SAS are taken as postulates in most treatments, or derived from rigid-motion arguments in transformation-based ones: given the matching parts, an explicit sequence of isometries maps one triangle onto the other.

The included-angle condition in SAS is essential. SSA fails because a circle of the given radius can meet the far ray in two points, producing two non-congruent triangles — the ambiguous case, which reappears in the Law of Sines in trigonometry. Whether zero, one or two triangles exist depends on the relationship between the given side and the altitude.

AAA also fails as a congruence criterion, since it fixes shape but not size. It is exactly the similarity criterion AA in disguise.

In practice, the marked diagram determines the route: three tick-marked sides suggest SSS; two sides with the angle between them suggest SAS. Shared sides (reflexive property) and vertical angles supply the missing third piece more often than any other pair of facts.