(not to scale)

The triangle ABC is inscribed within a square. If A=72º and B=84º, find the other angles in the figure (round to 2 decimal places if necessary).

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Sir,

Is this solution correct?

https://ibb.co/jyBnNZB

 

Edit:

 

https://ibb.co/jgncxfG
https://ibb.co/8D0fVKF

The last line has an error it should be

x = 114*sin72/(sin72+sin84)

And x turns out to be 55.726586093

rest can be easily calculated

Update 04-07-2020

https://ibb.co/TbyRQq8

https://ibb.co/jbBDPj3
by Level 8 User (30.1k points)
selected by

No, it isn’t. Remember that the triangle is inside a square, not a rectangle! Hint: you can use the Sine Rule to work out the relative lengths of the sides and then build up equations based on the fact that all sides of the square are equal.

Actually that was the first thing that I did,

Please have a look at the edit.
Also Sir,

What was wrong with this approach?

https://ibb.co/tbgfDZg

Edit: Okay I get it, my initial assumption was wrong.

Please validate my edited post above.

Thank You

No, unfortunately still not right. In my solution I did not find it necessary to draw any additional construction lines. It was all done through trigonometry. Yes, you’re right: given one angle you can derive all the others by simple arithmetic. There is only one solution. The clue is that the “bare” sides of the square can each be expressed in terms of one unknown angle. That gives you an equation, which needs to be simplified by using trig identities and a bit of manipulation to solve for the unknown angle. I have only glanced at your workings out so far, but I’ll take a closer look to see what assumptions you made. 

And what about the edit in original answer, did you look at that?

It is done using the law of sines.

I note that in equation (II) you have x instead of sin(x).

You don’t need the Sine Rule in a right triangle. So, for example, m=bsin(x) by definition of sine, and m=acos(x-24), which means that bsin(x)=acos(x-24). You have the right idea.

When you find x, check your answer by computing the lengths of the sides of the square after working out all the angles, and make sure that the sides are in fact equal.

Sir,

I posted a new solution, and I hope it is correct.

But I am unable to fulfil the new clause, I mean I am unable to find the length of the square, please give me a hint. And also please see if my solution is correct. As the primary requirement of the question was to find angles and I think I achieved that.

Thank you

If you think about it, you cannot find the size of the square because, no matter what the size of the square, the angles remain the same. Different size squares would all be similar figures. When I said check the sides of the square, I only meant check that they were the same size as a final check that you had the right solution.

And, yes, apart from a small error in rounding one of the angles, you have the right answer! And the method works. Well done for persisting!

Thank You Sir.

Can I have some more?

I’ll let you know if I find another meaty problem. There are certainly some I couldn’t solve myself amongst unanswered questions on this website. But that’s usually because I didn’t know enough about the subject matter to attempt a solution.

If you couldn't then what chance do I have on solving them.

You are wrong if the question is clear enough then I have seen you learning and answering them even if they weren't related to mathematics at all.

And I could easily search for lot of problems that I could try, but when you give it to me then I get motivated and give all that I have. I know if anyone have a slightest chance to teach mathematics to unintelligent student like me then he is you.

I am and will remain forever grateful to you. I see your efforts in helping others selflessly and it really touched me. If it were in my hands, I would have already given you field prize. Thank You Sir.
And yes for those who doesn't know, Sir stays awake very late at night only to help help-seekers like us.

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