the number of different triangles How many  different  triangles in which the measurement of the interior angles in degree is representing natural numbers in arithmetic sequence 

I believe you meant that if the angle of the triangle are $\left\{\alpha - k, \alpha, \alpha + k\right\}$ which are in arithmetic progression then the sum of these angles $=180^0$. Note, if three numbers are in AP then the the difference between the consecutive terms is constant.
$$\alpha - k + \alpha + \alpha + k = 180$$
So you have to just solve for $\alpha$ in the above equation
On the other hand, if you are interested in the fact that the angles are part of an Arithmetic sequence such that if the angles of a triangle are $\{\alpha,\beta,\gamma\}$ and 
$$\beta = \alpha + nk\,, \gamma = \alpha + mk$$
then
$$\alpha+\beta+\gamma = \alpha+\alpha+nk+\alpha+mk=180^o$$
$$3\alpha+k(n+m)=180^0$$
$$k(n+m)=3(180 - 3\alpha)$$
So if $k=3$ and $1 \le \alpha \le 59$ we will have the set of triangles whose angles are natural numbers and part of an Arithmetic Sequence
