Not sure if anything is already written on this, but if you're familiar with geometry methods, you could get at the 'line parts' which would be a point array (for lines that are not true curves) and use that to mimic the azimuth calculation by using a trigonometry calculation on the 1st 2 point coordinates in the array.
This should work fine for straight lines, and this is what this applies to, correct? I am asking because for lines with multiple vertices not falling along a straight line, this would make no sense. I'm not sure, but I have also worked with MBG (minimum bounding geometry) standard tools in the Toolbox, but think there is an added field calculated for you that could be used? You may have to perform some additional calculations, but that would save you from the details of accessing the geometry yourself.
EDIT: Can use the 'envelope' option, adding the MBG_Width and MBG_Length fields, from which you may derive inverse tangent to get the angle...only thing is you will have to track quadrant (+x,+y; -x,+y; -x,-y; +x,-y), i.e. direction by virtue of 'where' end coords plot relative to begin coords.
Enjoy,
Wayne