Let \(P\) be a polyhedron and \(E\) an equivalence relation on \(P\). This paper gives criteria for when \(P/E\) is a PL quotient space, i.e. when \(P/E\) admits a PL structure so that the quotient map \(q: P\to P/E\) is PL. Under the hypothesis that \(q\) is nondegenerate the author provides a criterion that is easily applicable and recovers some known triangulation results. For example, it is shown that if a finite group \(G\) acts on \(P\) through PL homeomorphisms, then \(P/G\) is a PL quotient space. Furthermore, interesting counterexamples to various weakenings of the author’s criteria are given.