You have not offended me, but rather surprised.

Of course it is a ruler-and-compass construction. Much in the spirit of what you've done. Pick a line and two arbitrary points on that line. At one point draw one angle, at the other the other angle. At the intersection, you get the third angle. You can proceed from there.

This is a mathematical tactics known as the reduction of the problem to a previously solved one. Of course you can combine the two steps.

Instead of going to the ASA construction, place on one of the constructed angles the given segment and draw a parallel, as you've done.

However, it is now quite clear that, in general, there are two solution. You can place your segment between angles 1 and 3 or between the angles 2 and 3. (1 and 2 are given and 3 is the one you just found.) Unless the given angles (1 and 2) are equal, there are two solutions. The SAA do not determine a unique triangle.