2006 AMC 10A Problem 19

Let the angles be $n - d,$ $n,$ $n + d.$ Their sum is $3n = 180,$ so $n = 60.$ The measures are distinct positive integers, so $d \ge 1,$ and $n - d \gt 0$ forces $d \lt 60.$ Thus $d \in \{1, 2, \ldots, 59\},$ giving $59$ non-similar triangles. Thus, the correct answer is C.