Congruence & Proof
Congruence Criteria
Using SSS, SAS, ASA, and AAS to establish triangle congruence.
How Little Information Actually Fixes a Triangle
Two triangles have the exact same three side lengths. Before reading on, predict: must they be exactly the same shape and size — congruent — or could two different-looking triangles share all three side lengths without being identical?
Formally, two figures are congruent exactly when some sequence of rigid motions maps one onto the other — the definition connects directly back to the last unit. For triangles specifically, it turns out you don't need to verify all six measurements (three sides and three angles) individually; certain smaller combinations are already enough to guarantee every other measurement matches too.
Definition — Triangle Congruence Criteria
Why does matching only three specific pieces lock in the whole triangle? Picture building a triangle physically from three given side lengths (SSS): once the three lengths are fixed, the triangle's shape has no freedom left to change — try to flex it, and at least one side length would have to change too. The same rigidity applies to two sides and their included angle (SAS): fixing that angle between two fixed-length sides completely determines where the third side has to close the triangle. A triangle, unlike a quadrilateral, is a rigid shape once enough of its measurements are pinned down.
Not every combination of three matching pieces works, though. SSA (two sides and a non-included angle) is famously insufficient — the same two side lengths and the same angle can sometimes be completed into two genuinely different triangles, an ambiguous case where the third side could swing to two different valid positions. And AAA (three angles) only fixes a triangle's shape, not its size — a small triangle and a much larger one can share all three angles while having completely different side lengths, which becomes the whole idea behind similarity, not congruence.
Worked Example — Identifying a Valid Congruence Criterion
Worked Example — Recognizing an Insufficient Combination
Tip
Common Mistakes
Treating SSA as a valid congruence criterion, by analogy with SAS.
SSA is not a valid criterion — the same two sides and a non-included angle can sometimes form two different triangles, so this combination doesn't guarantee congruence the way SAS does.
Assuming AAA proves triangles are congruent, since all three angles matching feels like 'everything matches.'
AAA only guarantees the same shape (similarity), never the same size — a triangle can be scaled up or down while keeping every angle identical, so matching angles alone says nothing about the actual side lengths.
Key Takeaways
- SSS, SAS, ASA, and AAS each guarantee triangle congruence by fixing enough measurements to eliminate any flexibility in the triangle's shape.
- SSA does not guarantee congruence, since it can sometimes describe two genuinely different triangles.
- AAA guarantees the same shape but not the same size, which is the foundation of similarity rather than congruence.
Summary
These four criteria let you prove two triangles congruent without measuring all six parts. The next lesson uses them as building blocks inside formal, step-by-step proofs.
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