# Thread: Can this problem be solved? Calculating angles of triangle

1. 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.

2. Originally Posted by Ariel4
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.
The page Denis referred to points out that, "[Langley's] problem ... has become known as the problem of "adventitious angles", because only for certain special combinations of angles is it possible for all the angles in the figure to be rational multiples of pi [i.e. of 180 degrees]." This same problem is called the "World's Second-Hardest Easy Geometry Problem" in the first link I gave you, which also attributes it to Langley. If this is not exactly the problem you are expected to solve, then there is no guarantee that it can be solved by the same methods -- and in fact we know that the answer is not of the same type.

The big question for you is, are you just saying this is SIMILAR to the problem you are asking about (as I said, too), or are you saying that this IS your problem, and the diagram you gave is wrong? It is really important that you state the problem correctly.

Again, you have been dribbling out information a little at a time, some of which seems to be incorrect. Does your teacher have the answer or not? Has he said the answer is a rational number of degrees, or not? Did he mention a circle or not? Perhaps you need to start over at the beginning and tell us the exact wording (and diagram) of the problem as given to you, plus exactly what he has said since. If he just said to find the angles, then you have solved the problem and should get the credit, regardless of what extra hurdles he puts in your way afterward. You can't be required to give a certain kind of proof; there are many. But the most important thing is that if the problem he gave you is not one of those that can be solved exactly by geometrical methods alone, then he can't expect you to do so.

3. Tomorrow I will ask professor again to tell me everything I need to know about this problem and I will write it here. Change of angles is intentional.As I wrote before, he said that those angles are correct.

4. Sorry for long time to reply but my professor was absent from school until friday (today).Apparently, he will get 6 months in jail because he:
-Didn't tell us correct angles
-When he realized that, he said that the corrected problem is to easy to solve
Anyway, I want to thank everyone who helped me!

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