jmc

Let $n-d$, $n$, and $n+d$ be the angles in the triangle. Then $$180=n-d+n+n+d=3n,\text{so}n=60.$$ 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.