三角形合同公理計算機

The Triangle Congruence Postulate Calculator answers a specific question: "given this set of measurements, which congruence postulate (SSS, SAS, ASA, AAS, or HL) applies?" It is the detector / identifier tool, complementary to the Congruent Triangle Calculator which proves congruence directly. This tutorial covers the 5 postulates, the decision logic for choosing which one applies, and the trap patterns (SSA, AAA) that look like they should work but don't.

The 5 valid congruence postulates

The decision tree

Given the measurements you have, work through this decision tree:

1. Are both triangles right triangles? If yes and you have hypotenuse + leg → use HL.
2. Do you have all 3 sides? If yes → use SSS.
3. Do you have 2 sides + 1 angle? Check if the angle is BETWEEN the two sides. If yes → use SAS. If no (SSA pattern) → NOT a valid postulate, see below.
4. Do you have 2 angles + 1 side? Check if the side is BETWEEN the two angles. If yes → use ASA. If no → use AAS (still valid).
5. Do you have only 3 angles? → NOT enough for congruence (only proves similarity).

Worked example 1 — recognizing SAS

Two triangles each have sides 7, 9, and the included angle 50°. Which postulate?

The 50° angle is between the two sides → SAS. The triangles are congruent.

Worked example 2 — recognizing ASA vs AAS

Two triangles each have angles 40°, 80°, and side 6 (where side 6 is between the two angles in both).

Side is between the two angles → ASA. Congruent.

If instead side 6 were opposite one of the angles (not between), it would be AAS — still congruent, different postulate name.

Worked example 3 — recognizing HL

Two right triangles each have hypotenuse 13 and one leg 5. Which postulate?

Both are right triangles + hypotenuse + leg match → HL. The other leg is forced to 12 by Pythagorean theorem (5-12-13 triple) so all six parts match.

The traps — SSA and AAA

SSA (Side-Side-Angle, non-included)

Two sides plus a non-included angle. This is the "ambiguous case" — the same SSA setup can fit zero, one, or two triangles. NOT a valid congruence postulate.

Exception: HL, which is SSA with a right angle. The 90° eliminates the ambiguity.

AAA (Angle-Angle-Angle)

Three matching angles. Only proves SIMILARITY, not congruence. The two triangles have the same shape but can be any size.

If you see AAA in a problem, you need at least one side match to upgrade from similar to congruent.

What if multiple postulates seem to apply?

Sometimes you have enough information for more than one postulate. For example: if you know all three sides AND all three angles, you can cite SSS, SAS, ASA, or AAS depending on which subset you emphasize. Pick the one that uses the fewest given items — usually SSS (simplest) or HL (if it's a right triangle).

Why these 5 postulates?

The 5 postulates cover all minimal sufficient combinations of triangle measurements:

• Specifying 3 sides → SSS
• Specifying 2 sides + 1 angle (included) → SAS
• Specifying 2 angles + 1 side → ASA or AAS (depending on inclusion)
• Right triangle: hypotenuse + 1 leg → HL (special)

Fewer than 3 elements isn't enough. More than 3 is redundant. The combinations that don't work (SSA, AAA) are the ambiguous patterns.

Real-world applications

• Surveying. Verifying that two triangulated land plots are congruent by measuring specific sides and angles.
• Engineering. Confirming that two manufactured triangular parts (truss components, frame supports) are identical.
• Quality control. Inspecting that produced parts match the specification.
• Computer graphics. Verifying that triangle meshes have the right congruence properties before rendering.

Common mistakes

• Citing SSA as a postulate. SSA is NOT a valid postulate (except HL). Two triangles with matching SSA may not be congruent.
• Confusing ASA with AAS. Both work, but the names are different. ASA = side IS between the two angles. AAS = side is NOT between (but opposite one of them).
• Forgetting HL only works for right triangles. Don't cite HL on non-right triangles.
• Treating AAA as congruence. Equal angles prove similarity only. Add at least one side for congruence.