How this calculator works
Each measurement case first determines a complete triangle. SSS uses the cosine rule, SAS uses the included angle, and ASA or AAS uses the angle sum followed by the sine rule. Sorting side lengths makes the comparison independent of vertex labelling. Matching all three lengths establishes congruence and supplies a vertex correspondence. Reflections count as congruent. Comparisons use a relative tolerance of one part in ten billion; this is a numerical comparison rather than a formal symbolic proof.
Formula
SSS congruence: corresponding a = d, b = e, c = f. SAS: c² = a²+b²-2ab cos C. AAS: c = a sin C/sin A with A+B+C = π.
Worked example
A triangle with sides 3 m, 4 m and 5 m is congruent to one labelled 5 m, 3 m and 4 m. Their vertices match A to B, B to C and C to A. A 6 m, 8 m, 10 m triangle is similar but has twice the side lengths, so it is not congruent.
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Frequently asked questions
Can the two triangles use different criteria?
Yes. For example, enter SSS for one and SAS for the other. Both are solved before their sides are compared.
Does a mirror image count?
Yes. Translation, rotation and reflection preserve side lengths and are allowed in congruence.
Why are AAA and SSA absent?
AAA does not fix size. SSA can describe two different triangles, so neither is a general congruence criterion.