How would you add a horizontal line without interference to the other edge? As shown on the image -
In edit mode press K, that will start the knife tool. Simply LMB click on the vertex you want the cut to start from, and left click again on the other edge. Snapping for the cut is enabled by default. If you want the new edge to be straight then activate angle constrain by pressing C. Once you like how the new cut is press Enter to finish the cut and exit the knife tool.