Let 0＜γ＜π be a fixed pythagorean angle. We study the abelian group H_{r} of primitive integral triangles (a,b,c) for which the angle opposite side c is γ. Addition in H_{r } is defined by adding the anglesβopposite side b and modding out by π-γ. The only H_{r} for which the structure is known is H_{π}_{/}_{2}, which is free abelian. We prove that for generalγ, H_{r} has an element of order two iff 2(1-