I am having trouble cutting the angles on the stairway. How do I find the angles?

I assume you are using a mitre saw to cut the chair rail.

I've done this a bunch of ways:
1) use an adjustable square and lock it at the angle you want. http://www.woodville.co.nz/images/dcp00272.jpg
Put the square on your mitre saw to measure the angle and then cut at half that angle.

2) Use a speed square to measure the angle. http://images.oldhouseweb.com/storie...eedsquare1.jpg

3) trial and error with smaller pieces till you get it right! You'll probably find that the first two methods will get you close and then you'll have to fiddle a bit to get it perfect.

Without doing this myself now to make sure, you may find you have to add or subtract 90 degrees to get the proper angle before you divide by two for the mitre joint angle.

A stairway should be fairly consistent on the rise and run, without getting into degrees and stuff, ill tell you an easy way to do this. First measure up 36" from the floor this is the top of the chairrail. chaulk a line or draw a level line useing a level. also do this at the top of your stairs. now measure from the very front of your stair tread( right on the edge of the nosing) the rounded part. measure up there 36" using the level so its straight up, just make a small mark there. You want to make a mark on the top step and the bottom step. Once you do that chalk that line. Now what you should see on your wall is just like the diagram you sent use to look at, follow me? OK next take a piece of chairrail and hold it with the line you just made... start at the bottom is fine, let it run past your mark 4 or 5"
now the line that goes up the steps. do the same thing there and let it run past 4 or 5". OK as you look at this you see the intersecting points... thats your angle. Simply cut a piece long make a mark on the top and now you can mark the bottom as well, if you don't have a speed square to find the degree just cut it a few time in the miter saw till your right on those marks you made. If I had to guess your degree may be like a 19 or 20 degree. Let me know if this worked for you, good luck

Glavine can do my house anytime.

You can use a level and a protractor to find the angle. Make a level mark, then measure up at the step as glavine said on two steps, make a line to each, then you will have an angle. Subtract the angle from 180 degrees and that is the angle you cut.

If there is a window at the top of the stairs you may want to use the trim under the sill as the height of the chair rail.. Start here.

If not start about 36" height. Draw a chalk line the length of the rail.

Use a "angle" to determine the angle of each joint. Divide the angle in half and cut both with a mitre saw.
e.g. If your angle is 42 degrees cut 21 degrees and 21 degrees.

Pre drill nail holes so you don't split any railing. Caulk joints, top and bottom of railing.

If you do not end at a wall or door frame then mitre the end with the second piece being the same width as the chair rail