I want to transpose an angle about a 45 degree line. Imagine a right angle called ABC with A on the horizontal. Now imagine an acute angle ABD. I want to construct an angle ABE such that angle EBC is congruent to ABD and ABE and ABD are complementary.

I think the way to do this is to position a compass at B and strike an arc across the whole quadrant. Position one compass point where the arc crosses BD and the other where it crosses AB. Move the compass to where it crosses BC and strike an arc across the first arc. Call this E. Connect E to B and you have your transposed angle.

The issue is that I am not sure it’s allowed to adjust the compass to the distance between the point where the arc crosses BD and the one where it crosses AB.

For context, I am trying to figure out how to strike a line on a board using only a speed square and a bevel gauge that is the complement of another angle.

So, it does not sound like there is any theoretical problem. Have you tried it? What is the issue? I mean, you say there is an issue adjusting the compass— is it not precise enough? Is the angle very small?

Also, it seems like if you have a gauge set for, say, 12 degrees, and you have a 90-degree angle, then you can measure those 12 degrees starting from either side of the angle.

In the classic ancient Greek meaning of “geometric construction”, using the compass as a distance measuring tool akin to a calipers or dividers was forbidden. The adjustment you suggest amounts to that. So unkosher. Likewise measuring distance by using the straightedge as a ruler is unkosher.

In getting a job done at work, well, practice overcomes theory all day every day.

IANA carpenter, so I’m not competent to offer the best way to use your particular tools. But is sure smells to amateur me like @Retzbu_Tox just above a) knows what he’s talking about and b) has a (the?) right answer.

You don’t have to measure anything— there is no reference scale, but you can say when two lengths are equal, or construct a segment of a given length at a given point, which was probably not super forbidden since Euclid does just that in Proposition 2.

Construct a right angle DBF (Trivial exercise ) CBF is congruent to ABD

Draw an arc from A to F centered on B call it B-AF
Draw another arc from C that goes through F call it arc C-F
Where arc C-F intersects the first arc B-AF on the other side of C is E such that the angles are all congruent - ABD, CBF, CBE

Euclid prohibited the use of a compass for directly measuring just because he could. He very quickly showed that even if you couldn’t do it directly, you could do it indirectly. Once you have that proof, you can proceed in just the same way as if you could pick up the compass and keep the same setting on it.

Other mathematicians did make fun of this, but it’s the best way to proceed with proofs: Assume as little as you can get away with, to prove as much as you can. This also means that, if your proofs are translated to some other context, more of them can survive the move.