Side-side-angle triangle congruence axiom and the complete quadrilaterals

Abstract

Among the triangle congruence axioms, the side-side-angle (SsA) axiom states that two triangles are congruent if and only if two pairs of corresponding sides and the angles opposite the longer sides are equal. We construct two triangle sequences in which the items satisfy a modified condition. We require that the opposite angles of the shorter sides be equal. The locus of the intersection points of other sides of triangles is derived to be a hyperbola, and in a generalized form defined by a complete quadrilateral, it is a conic section.