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

1. ## Can this problem be solved? Calculating angles of triangle

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! kec.jpg

2. I'm pretty sure extra credit requires extra effort, not a handout. Let's see your best efforts.

3. Originally Posted by Denis

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 :/

4. Originally Posted by tkhunny
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.

5. 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.

6. Originally Posted by Denis
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'.

7. Originally Posted by Ariel4
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, ...)?

8. 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!

9. Originally Posted by Ariel4
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.

10. Originally Posted by Dr.Peterson
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"?

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