Is it possible to find a 25-degree angle using a ruler and compass only? If so, how is this done?

The answer is no.

The only integer numbers of degrees which are constructible

are those which are multiples of 3°.

http://planetmath.org/encyclopedia/ConstructibleAn...

I've given this question quite a bit of thought (and will probably give it more) but here's where I am so far.

Since I don't know how to create a 25 degree angle directly, I've been looking at it from the standpoint of altering a known angle to end up with one. Bisecting any angle is easy, and creating a 60 degree angle is fairly simple (equilateral triangle--all sides equal and easy to create with a compass and straight edge). What I've been looking into is trisecting an angle. If a 60 degree angle were trisected, (3) 20 degree angles would be created (IF THAT CAN BE DONE). Anyway, then one 20 degree angle would be bisected twice. That would create a 5 degree angle, and if it is next to one of the 20 degree angles. Viola!!

Big hangup is trisecting an angle. I've enclosed a couple of promising sources.

Tangent is the rise over the run, for 25 degrees this number is about .4663.... but not exactly .4663 ...

On a graph:

Draw a dot at 0,0 and a dot at 1,0.

Draw a dot at 1, tan 25 (about .4663 or so)

Connect the dots.

The angle inside the triangle at the left will be 25 degrees. The angle at the right will be 90 degrees. The angle at the top will be 65.

See? easy peasy.