How Many Triangles?

There are $5$ lines with slope $0$ , $5$ lines of the same positive slope, and $5$ lines of the same negative slope. Any triangle must include exactly one line of each slope. Now out of the $25$ intersections between the lines with positive and negative slopes, $6$ intersect without a third common intersection with a horizontal line and $19$ do. So if we choose one of the $6$ pairs of lines without a third common intersection for a triangle, then we have a choice of $5$ horizontal lines for a third side otherwise there are only $4$ horizontal lines to choose from. Thus the answer is $6(5)+19(4)=\boxed{106}$ .