2006 AMC 10A Problem 19

Question

How many non-similar triangles have angles whose degree measures are distinct positive integers in arithmetic progression?

0

1

59

89

178

If the angles are in arithmetic progression, the middle angle is

60^\circ

The common difference

d

satisfies

1 \le d \le 59

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.

Problem 19 in other years