Question

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

$0$

$1$

$59$

$89$

$178$

Answer: C

Concepts:angle sum arithmetic sequence counting integers in a range

Difficulty rating: 1630

Small Hint:

If the angles are in arithmetic progression, the middle angle is $60^\circ$

Big Hint:

The common difference $d$ satisfies $1 \le d \le 59$

Solution:

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.

2006 AMC 10A Problem 19

Answer: C

Difficulty rating: 1630

Small Hint:

Big Hint:

Solution:

Problem 19 in Other Years