So to be completely honest I'm not sure if even my teacher know what he wants from us.When it comes to circle, we need to apparently then construct one and use inscribed angles to get somewhere I pointed out that the solutions to this problem are not nice numbers at all as I checked it on GeoGebra but all he said is that you can get "nice" and not approximate angles without usage of calculator and only with things you have learned so far.Maybe he meant to we work out with angles all along (not to calculate sin70 for example, just to leave it there)and then at last step to do something with the solution.So to sum it up: When he said nice and not approximately I think he meant that we don't have to work with long decimal numbers except in the last step.To solve this prob, we need to draw few more lines in the triangle and work out with proving that most triangles here are isosceles and also use peripheral(inscribed) angles in some way.

As I saw in Peterson's links some of the solutions were solved by adding more lines to it and thru them calculating angles, but those lines are added specifically for that case of angles(triangle 80-80-20 with 2 known angles 50-60 degrees).

I found out that this problem is called Langley's problem.The main solution to this (as I saw online) is to add one line that will separate angle A (bottom left angle,80degrees) into one of 60 and 20 degrees.But that try also didn't work out for me, cuz in my case, I can't get isosceles triangles that you are supposed to get.

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