How to prove the four determination theorems of triangles such as SSS, SAS, AAS, and ASA?

Proving the four triangle congruence theorems—SSS, SAS, ASA, and AAS—fundamentally relies on establishing that given their specific sets of congruent corresponding parts, two triangles must be identical in all respects, including their third angles and remaining sides. The logical foundation for these proofs is not axiomatic but is built upon more primitive geometric concepts, most critically the rigid motion of figures in the plane. The core idea is that if one triangle can be superimposed perfectly onto another through a sequence of basic isometries (translations, rotations, and reflections), then the triangles are congruent. Therefore, each proof strategy involves strategically placing the two triangles in a position that facilitates this comparison, typically by mapping one vertex and one side onto their counterparts, and then arguing from the given postulates and theorems that this forces the remaining vertices to coincide.

For the Side-Angle-Side (SAS) postulate, it is often treated as the foundational axiom for congruence in many modern axiomatic systems, such as Hilbert's. If it is taken as a postulate, it requires no proof. In frameworks where it is a theorem, the proof proceeds by placing the two triangles so that the congruent angles and their included sides coincide. Given that two sides and the contained angle are fixed, the endpoints of the remaining sides are determined uniquely, forcing the third vertices to occupy the same point in space, thus proving congruence. The Angle-Side-Angle (ASA) theorem is then typically derived from SAS. By knowing two angles and the included side, the sum of angles in a triangle forces the third angle to be congruent as well. This effectively gives an angle, the included side, and then the next angle, which can be reinterpreted as a configuration where SAS applies, either directly or after a slight reordering of the parts.

The Side-Side-Side (SSS) theorem is more subtle because three given sides constrain the triangle's shape without an initial angle. A common proof uses a construction argument: place the two triangles with one congruent side aligned. Then construct circles centered at each endpoint of this base with radii equal to the lengths of the other two sides. The third vertex of each triangle must lie at the intersection of these two circles. Under the assumption that the geometry is Euclidean and that the triangle inequality is satisfied, the two circles intersect in exactly two points, which are reflections across the base line. Since reflection is an isometry, both possible triangles are congruent, proving that all triangles with those three side lengths are congruent to each other. The Angle-Angle-Side (AAS) theorem is directly reducible to ASA. Given two angles and a non-included side, the fact that the sum of interior angles is always 180 degrees allows the immediate calculation of the third angle. The side is now positioned between the two known angles—the original given side and the computed angle adjacent to it—which satisfies the ASA condition, thereby proving congruence.

The practical implication of these theorems extends beyond mere proof; they establish the minimal conditions for triangle uniqueness in Euclidean geometry. They are the essential tools for all subsequent geometric deduction, from proving properties of parallelograms to the similarity of figures. It is critical to note that the order matters: for instance, SSA (Angle-Side-Side) is not a valid congruence theorem except in the specific case of a right triangle (the HL theorem), because the given conditions can lead to two distinct, non-congruent triangles. The proofs collectively demonstrate that a triangle is rigidly determined when the given data fixes all its vertices unambiguously relative to each other, which these four combinations reliably do.