Draw the desired line - then it is available for inspection, zoom in, tweak ... whatever.
When you are happy with placement, use the Knife tool. Cut at the intersections - notice the Knife will automatically snap to an intersection. So - you make two (or more...) cuts. Now you have the two pieces of the cut graphic and the new closing segment (doesn't need to be a line). Select the desired elements and perform Convert to Welded Bezier. If both elements need "closing" duplicate-in-place the cut-closing segment so you have one for each half.
If the cut-intersections are near a vertex - use Option key with the knife to favor "intersection" over "node delete".
If the combined result has more control points than are needed - use Graphic Details to re-format the desired segments.
Dave