The purpose of this question is trying to know originally how did counting in integer degrees angles from (one degree to $360$ degrees) being adopted basically in geometry, despite the impossibility or the great difficulty of constructing exactly (without any approximation methods) those many integer degrees angles that aren't divisible by $3$,
So, I thought that there may be a very good unknown historical reasons (mainly by the ancient Babilions or Egyptians) to adopt such systems instead of using the exactly constructible angles defined essentially by the full angle of twice $\pi$
Thanking your opinions
Editing: October 4, 2018
No, the point of my question is quite different from the answers provided that well-explain the historical details about the issue, for the basic reason that many angels are impossible construction, where this strictly includes a subset of majority (or exactly two thirds) of integer degree angles of the form ($3n +/- 1$), where $n$ is a natural number
The other more important point about the true reason of the impossibility of constructing exactly certain angels, which was well proven by Wantzel in 1837, regarding the angle $\pi/9 = 20$ degrees, where despite the fact of the proof, people (including the proofers) still think that such angle exists but impossible to construct
And finally the most important point that might seem shocking and not at all acceptable by the mainstream academic mathematicians (for the first while), which states that such angles as $\pi/9 = 20$ degrees, doesn’t exist at all, same like all those integer degrees angles of the above-described form $3n +/- 1$,
So, to say why the angle of $20$ degrees is impossible construction is simply because it doesn’t exist at all Otherwise, how can an angle or a real number exist but can’t be exactly constructed? Wonder! To explain this simple fact, it is impossible existence for a triangle with exactly three known sides to have exactly an angle of $20$ degrees, or $19$ degrees or $22$ degrees, or generally any integer degrees angle which isn’t divisible by $3$, of course one shouldn’t use any approximation methods and concludes or claims the exact construction by means of limits or convergence or cuts or else, since all those methods are basically and strictly with endless number of steps (theoretically unfinished operation) So, the hope lies upon the mathematicians to give themselves a little chance not to miss this very important and new point of view
However, the whole topic is deeply related to real number concept that had been extended wrongly beyond the real positive constructible numbers (which are the only true real existing numbers), where this had been well-proved and were also published publically in other forums and also some here on SE, but were deleted from SE sections by moderators since they so, unfortunately, contradict strictly many common beliefs among mathematicians
EDIT: NOV. 25th, 2018 Adding reference: https://www.quora.com/Do-most-of-the-named-angles-in-mathematics-truly-exist
I thought of not adding my content here about my earlier claims regarding the absolute non-existence of majority of well-known angles in current and old mathematics as well since truly most likely it would be immediately deleted as were done to many of my other deleted claims here and generally at SE
But the fact which is more important than all of us, for sure