Can this problem be solved? Calculating angles of triangle

Ariel4

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So my teacher gave this math problem to us so we can solve it, he said that it is very hard even for him to do it.The person who solves it till tomorrow will get 2x A (double A's, double 5).I can't do it, i tried everything but it just seems impossible. I'm 17 y.o. , third grade in high school.I hope someone can help me with this.


-->We need to find those two angles that are marked on the picture, everything we know is marked and numbers on the picture are angles in degrees.Can you solve this and tell me how you get those answers? I need this ASAP. Thanks in advance!
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I'm pretty sure extra credit requires extra effort, not a handout. Let's see your best efforts.
 
Can you answer following:

1: what kind of triangle is ABC?

2: angle ADB = how many degrees?

3: angle AOE = how many degrees?

1.Isosceles triangle (with 2 equal sides)
2.70 degrees
3.130 degrees
But I don't see how these answers help in any way :/
 
I'm pretty sure extra credit requires extra effort, not a handout. Let's see your best efforts.
After 5 days of trying and consulting with another math professor both of us have no clue how to do this.I'm not even sure that this is possible to do.
 
I haven't spent time with this, but it reminds me of some similar problems I have seen, and MAY be solvable by the same methods. See here, here, and here.

On the other hand, I drew it with GeoGebra, and the answers are not nice numbers as far as I can see; it is possible that you have a garbled version that can't be solved so easily. I'll see what I can do with it.
 
You got me hooked on that darn problem!

Here are the steps I took to solve it:

Let AB = 1

Calculate AF (Sine Law)
AF = SIN(20) / SIN(130) = ~.446
(not showing complete solution for the others;
you do the work!)

Calculate BF (Sine Law)

Calculate EF (Sine Law)
(notice that triangle AEF is isosceles)

Calculate DF (Sine Law)

Calculate DE (Cosine Law)

Calculate angle DEF (Sine Law)

angle EDF = 180 - 130 - angle DEF

Perhaps Doc Peterson can do it in lesser steps...
Denis thank you so much for this,i solved it and got 18 and 32 degrees as solution(approximately).I gave this answer to my math teacher and he said that this is not the way i was suppose to do it. As he said apprently i don't need to use calculator and as a hint he told me to use peripheral angles and as i understood him ,try to represent angles with another angle(don't know right term for this in english but here is example. if sin130 => sin(150-20) etc and than to simplify those angles furhter more,sin and cos of double angle etc...).
And Peterson i checked your sites also,but as far i can see it is not done in same way as my proffesor imagined(with peripheral angles).Thanks for sharing those sites tho'.
 
Denis thank you so much for this,i solved it and got 18 and 32 degrees as solution(approximately).I gave this answer to my math teacher and he said that this is not the way i was suppose to do it. As he said apprently i don't need to use calculator and as a hint he told me to use peripheral angles and as i understood him ,try to represent angles with another angle(don't know right term for this in english but here is example. if sin130 => sin(150-20) etc and than to simplify those angles furhter more,sin and cos of double angle etc...).
And Peterson i checked your sites also,but as far i can see it is not done in same way as my proffesor imagined(with peripheral angles).Thanks for sharing those sites tho'.

Your answers are approximately what GeoGebra gave me, so you are presumably right; the program probably finds its answers using coordinate geometry, which is yet another possible approach.

One of the sites I referred you to gives 12 different approaches, some geometrical, some trigonometric. There are many ways to solve any problem like this; if the teacher wanted a particular way, or a particular form of answer, he should have specified that.

Unfortunately, I don't know what you mean by "peripheral angles", and you haven't given enough information to guess what specific approach he used.

I myself was expecting (because of the similar problems I pointed out) that it might be able to be done exactly as integer or rational degrees, by geometrical methods rather than trigonometry. I think you are saying that his method gives exact values by using trigonometry with special angles and identities, so that you would get a radical solution. If so, then you might be able to use the same method you did, but with such exact values rather than decimals. However, since trig functions of 20 degrees are not expressible in terms of square roots (because that angle is not constructible), the best I can expect is that somehow values like sin(20) could end up canceling out, leaving you something relatively simple. Do you know at least what kind of answer he got (rational, radical, decimal, ...)?
 
Oh, my bad, when I wrote peripheral, the right term for that is inscribed angle. As he said, it is very hard to get to a solution, and you need to use methods which I mentioned in the last reply.

The answer is something nice and it is not approximate angle. Also, I think he said something about constructing and adding more "lines" inside of triangle. Definitely a hard problem, and even harder to do it as my professor imagined in the first place.

And, Dr.Peterson, I have now just seen that those 12 solutions are really 12 solutions. I though they were 1 solution split into 12 parts (my bad). I will check them all as soon as I find free time. Thank you!
 
Oh, my bad, when I wrote peripheral, the right term for that is inscribed angle. As he said, it is very hard to get to a solution, and you need to use methods which I mentioned in the last reply.

The answer is something nice and it is not approximate angle. Also, I think he said something about constructing and adding more "lines" inside of triangle. Definitely a hard problem, and even harder to do it as my professor imagined in the first place.

And, Dr.Peterson, I have now just seen that those 12 solutions are really 12 solutions. I though they were 1 solution split into 12 parts (my bad). I will check them all as soon as I find free time. Thank you!

I had searched for the term "peripheral angle" and found that it seems to be used sometimes for inscribed angles (in a circle). But in the figure itself there are no circles; and at least one of the proofs in my references uses inscribed angles, but you didn't mention that as being similar to what you were told to do, so I had to question that. You also say that you were told to construct lines inside the triangle, not outside of it and in a circle. Have you told us everything your teacher said?

So you are saying that the problem is to find the exact value of each of the two angles, presumably either as a rational number or using simple square roots. Is that right? (I wish the problem were stated completely from the start.) That does at least confirm my hope that a purely geometrical method should be possible, though you indicated that trig is expected, though my last comments indicate why I expect your trig approach to be particularly tricky.

I'll give it another try.
 
The answer is something nice and it is not approximate angle.
So you are saying that the problem is to find the exact value of each of the two angles, presumably either as a rational number or using simple square roots. Is that right? (I wish the problem were stated completely from the start.) That does at least confirm my hope that a purely geometrical method should be possible, though you indicated that trig is expected, though my last comments indicate why I expect your trig approach to be particularly tricky.

When I wrote that, I was confusing the fact that exact values of trig functions can be radical expressions, with what you are actually looking for, which are angles -- exact values of inverse trig functions, ultimately, and in degrees. It doesn't really make any sense to expect exact values other than rational numbers of degrees (like 7 1/2 degrees, for example). And the numbers we all got don't look anything like rational numbers.

So, please be very specific: in what sense did your teacher say the answers are "nice" and "not approximate"?
 
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 :confused: 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.
 
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 :confused: 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.
 
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.
 
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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