One nice way to note the angles in these triangles is to form window patterns based on them - these are shapes made from overlapping pieces of paper that have been rotated according to a rule. The term window pattern comes from William Gibbs - so named because if you put them up in a window, the light shining through the different layers of paper reveals additional patterns and shapes.
Here's one example of the special-triangle-window-pattern process. Start with an A4 or similarly proportioned rectangle, and find the midpoint of one of the shorter sides (by folding the paper, for example).
These are nice patterns, but they don't actually use the special properties of A4 (you could do a similar thing with square or letter paper). A little more complicated placing of one rectangle over the other can allow you to create a right triangle with one leg equal to 1 and the other equal to sqrt(2)-1. This is not one of your "standard" special triangles, but it is special in that it allows you to calculate exact values of certain angles (which angles, we'll find out when we complete our pattern).
Here's what the placement looked like that constructed this triangle. I'm afraid that text instructions for the placement would be just too much for this post - maybe you can figure out how it is done from the diagram :).