It is shown that in the Euclidean plane, if two triangles share a side-length and an angular measure (resp., share a multi-set of angular measures) and have the same area, then the triangles are congruent. If M >  0, 0 < α < π and a > 0, a necessary and sufficient condition is given for the existence of a triangle Δ = ΔABC (that is necessarily unique up to congruence) such that the area of Δ is M, the measure of  is α, and the length of the side BCis a. A possibly novel proof is given that if two (Euclidean) triangles satisfy the same “SSA” data for the celebrated “Ambiguous Case” but have incongruent “included angles,” then the triangles are incongruent (even after any permutation of one triangle’s vertices). As the only prerequisites for this note are elementary analytic geometry and trigonometry, these results could find use as enrichment material in a precalculus course.

Euclidean plane, triangle, area, side-length, congruence, trigonometry, ambiguous case, analytic geometry.