For a given positive integer k [not equal] 4, let "P[subscript k,n]" denote the "n"-th "k"-gonal number. We study "k"-gonal triples ("a", "b", "c") satisfying P[subscript k,a] + P[subscript k,b] = P[subscript k,c]. A "k"-gonal triple corresponds to a rational point on the rectangular hyperboloid x[squared] + y[squared] = z[squared] + 1. The simple observation that this is a doubly ruled surface leads to an easy, complete determination of "k"-gonal triples. The main result is expressed in terms of primitive Pythagorean triples. We also establish a correspondence between (2"h" + 1)-gonal triples and 4"h"-gonal triples. (Contains 6 tables.)