How many non-similar triangles have angles whose degree measures are distinct positive integers in  arithmetic progression?
Let $n-d$, $n$, and $n+d$ be the angles in the triangle. Then \[
180 = n-d+n+n+d= 3n, \quad \text{so} \quad n=60.
\] Because the sum of the degree measures of two angles of a triangle is less than 180, we have $$180 > n + (n+d) = 120 + d,$$ which implies that $0<d<60$.

There are $\boxed{59}$ triangles   with this property.