You can still use Smashboxed's idea, then bisect the angle, to get the angle you need for cutting the wood.
How do you bisect the angle? You use a drafting compass or dividers. From the intersection of the two sides of your original angle, place one side of the compass at the intersection then mark a distance along one side. Then without changing the compass, do the same on the other side. Then put one end of the compass on the first mark you made and trace an arc in between the two sides (about half of the original angle). Then do the same for the other side. Locate where the arcs intersect each other. Now draw a straight line between the original intersection and the intersection of the arcs. You just bisected the original angle (split it exactly in half).
If this is too much work for you (it is a lot easier than it sounds) then you could just go to Home Depot or a similar store and buy an adjustable plastic protractor which is like the homemade one that Smashboxed is describing. I think I paid about $8-10 for the one I have.