Hi Stu,
Single point and knife—so you have used it already. :o) “Only Visible”—Yes, I know that feeling, happens.
BTW, the Pen Tool has an option as well for this:
Shift+clicking on an edge splits the edge by adding a point that can initially be moved along the length of the edge as a preview. Release the Shift key to position the point at any location. Click on an edge to affix the point.
To have an edge extruded with a specified angle, sounds more like the Extrude Tool to me.
With the Polygon Tool, the Snapping functions or Work-planes might be an idea.
I certainly suggest Donavan Keith’s series:
http://www.cineversity.com/vidplaylist/polygon_pen/polygon_pen_part_01
I just watched it, to not miss an option, as I couldn’t recall any directional option (as in Extrude>Edge Angle). No wonder I couldn’t, there is obviously none. ;o)—which is also confirmed checking the Help Content.
I hope that help a little bit.
All the best