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 angles β opposite 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- cosγ) is a rational square, and it has elements of order three iff the cubic (2cosγ)x³-3x²+1=0 has a rational solution 0＜x＜1. This shows that the set of values ofγ for which H_r has two-torsion is dense in [0, π], and similarly for three-torsion. We also show that there is at most one copy of either Z₂ or Z₃ in H_r. Finally, we give some examples of higher order torsion elements in H_r.