Formula and Method for Triangle Congruence
Two triangles are congruent when every corresponding side and angle is exactly equal — the triangles are the same size and shape, even if one is rotated, reflected, or translated relative to the other. You almost never need to check all six measurements: geometry guarantees congruence once a specific, smaller set of corresponding parts match. This calculator lets you pick one of the five standard congruence postulates — SSS, SAS, ASA, AAS, or HL — enter the required measurements for each triangle, and see whether they match.
How the calculation works
For each triangle, the calculator solves the remaining sides and angles from the values you enter, using the Law of Cosines (c² = a² + b² − 2ab·cos C) to find a missing side or angle from two sides and an angle, and the Law of Sines (a / sin A = b / sin B = c / sin C) to find missing parts once two angles are known. For SSS it also checks the triangle inequality (the sum of any two sides must exceed the third) to confirm the three lengths actually form a triangle. It then compares Triangle 1's entered values to Triangle 2's entered values, slot by slot; if every corresponding value matches (within a small rounding tolerance), the triangles are congruent by that postulate.
Common mistakes
- Using SSA: two sides and a non-included angle is not a valid congruence test — the same three values can describe two different triangles (the "ambiguous case"). Only SAS, where the angle sits between the two given sides, works.
- Treating AAA as congruence: matching all three angles only proves the triangles are similar (same shape), not congruent — they can still be different sizes.
- Mismatched correspondence: the postulates only work when the matched parts correspond to the same vertices in the same order (e.g., ∠A with ∠A, side a with side a). Comparing the right numbers to the wrong parts invalidates the proof even if the digits happen to match.
- Degrees vs. radians: all angle inputs and outputs on this page are in degrees, not radians.
Real-world applications
- Structural engineering and truss design rely on triangle rigidity: a triangle with fixed side lengths (SSS) cannot deform, which is why trusses and bridges are triangulated.
- Surveying and navigation use triangulation (ASA/AAS with measured angles) to fix unknown distances and positions.
- Manufacturing and CAD tolerancing use congruence checks to confirm that duplicated parts are dimensionally identical.
- Geometry proofs and coursework use SSS, SAS, ASA, AAS, and HL as the standard toolkit for proving two triangles congruent.