What does it mean for an edge to be "on the left side of the triangle"? I thought of a couple of different ways an edge might be considered "on the left":
1. Edge's midpoint is left of the triangle centroid
2. If there is only 1 left-most point, then the two edges that have the left-most point would be left edges
3. If there is only 1 right-most point, then the edge which doesn't have the right-most point is left edge
In short. That it is pointing downward on screen. (Assuming that your triangles follow a consistent counter-clockwise winding on screen.)
Here's a good reference. Look for the section call "fill rules".