I can not match the chair rail cut from the stair way which is approximatly at a 50 deg. Angle up the stair way to the top of hallway on an outside corner. The piece of chair rail that comes up from the stair way is wider after I cut it because of the angle that I have to have coming up the stairs than the piece that intersects at the outside corner from the hallway up stairs, just mitering it at a 45 deg does not match.

Are you saying that the piece going up the wall ends at a corner. If so, hold the piece going up the stairs back about 6", go horizontal and intersect the outside corner horizontally.

If your stair way is a standard rise and run, your angles should be around 20 to 25 degrees. Id start with 25 and see what that looks like, that should be close enough to start making some adjustments

Hold The Piece Back The Same Distance As The Top Step And Go Horizontal