I mean... hmmm... I presume they measure the trisected angle with a protractor, but, how do they know the protractor is accurate, and that the compass drew a perfect arch, and that the ruler is a perfect straight edge. Not to mention that a pencil cannot be sharpened to an infinitesimal point (and in any case, we can't see infinitesimal), that human eyes and hands cannot place the pencil, ruler, or compass in the exact spot......
They didn't bother with any of that. They were using "ideal" instruments.
For the most part, Ancient Greek geometry was an exercise for the mind. That it also had practical consequences was something they were willing to live with.