One very useful function here is to use Polar instead of Ortho, you can then select "Relative to last segment" instead of "Absolute" in the settings dialog. now if you start drawing by Osnapping to your arbitrary line then its direction is a temporary 0 degrees.
setting the UCS to this line is probably best if you have a fair bit more to draw or it is at a strange angle to your base line.
The other option is to draw your new stuff on the default alignment then move and rotate it into place. Investigate the use of the reference option for rotate, it will let you align it with your arbitrary line before you rotate a second time by your fixed amount. Also check out the ALIGN command, designed for 3d but great in 2D as well (simply press return when asked for the 3rd reference point