Determine remaining 3 angles of trapezium with 1 right angle.

I know the length of all 4 sides of a trapezium
I know that one of the angles is 90 degrees.
None of the sides are parallel to the others.

How can I determine the remaining 3 angels?
AD1080Asked:
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ste5anSenior DeveloperCommented:
As you know all sides length, you can determine the length of the diagonals. With these lengths you can use the trigonometric functions to determine the angles.
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d-glitchCommented:
I have an Excel spread sheet that solves quadrilaterals.  

You can get it here:  https://www.experts-exchange.com/questions/29061368/Irregular-rectangular-area-calculation.html
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AD1080Author Commented:
Thanks for the input.  I lack a reasonable education in trig.   Could you provide the formula you have in mind?
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aburrCommented:
You cannot get unique angles without some additional information.
Can you tell me which lengths make up the right angle?
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Paul SauvéRetiredCommented:
trapezium, by definition:
  1. (British English) (North American English trapezoid) a flat shape with four straight sides, one pair of opposite sides being parallel and the other pair not parallel
  2. (North American English) (British English trapezoid) a flat shape with four straight sides, none of which are parallel
so are you using the North American or the British definition?
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phoffric\Commented:
He is using north American since the op states that none of the sides are parallel to each other.
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phoffric\Commented:
Assuming that you are able to draw your trapezium with the given lengths, then there is one unique solution for a convex trapezium and another unique solution for a concave trapezium.

 For the convex case, draw the hypotenuse of the two sides that have the right angle. This gives you a right triangle and you can find those angles easily. Here is a link that shows you how to do that:
https://www.mathsisfun.com/algebra/trig-finding-angle-right-triangle.html

 That hypotenuse line is a diagonal of your trapezium and it's formed two triangles one being the right triangle that you just worked on. Now you know the length of the hypotenuse using Pythagorean theorem. Now you are left with another triangle where you know the lengths of the three sides where one of them is the hypotenuse that you just calculated.

 You can use the law of cosines to compute the three angles of the triangle. Here is a link that explains a lot of cosines:
https://en.m.wikipedia.org/wiki/Law_of_cosines
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phoffric\Commented:
>>  I lack a reasonable education in trig.
Ok, here is a short video to show you the basics of the trig functions using a right triangle.

https://m.youtube.com/watch?v=X5uFqpypDy4
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phoffric\Commented:
Now that you have become a little familiar with the basic trig functions , here is a video that shows you how to compute the angles of a right triangle:

https://m.youtube.com/watch?v=j4O3fpC4HFg
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phoffric\Commented:
For the convex case you still have another triangle to compute the angles . Here is a video that gives an example of using the law of cosines. Shows how to find three angles of a triangle given known three side Lengths:
https://m.youtube.com/watch?v=xRnhvFCGrIk
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phoffric\Commented:
Finally there is the case of the concave trapezium to consider, but this case can lead to some strange figures including something like a figure 8 as well as something like a quadrilateral that looks like it only has two sides. That's why I assumed the easier case where you have a convex trapezium. If you have a concave trapezium, I suggest addressing this in a separate question and giving the line lengths and draw a sketch of the figure so as to keep the problem solution unique.
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AD1080Author Commented:
Thanks a lot!
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aburrCommented:
How about a solution using no math at all?
All you need is a protractor capable of measuring your angles with the precision required for your problem
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aburrCommented:
The difficulty is that the responders were trying to help the questioner get the answer by working out steps suggested without just giving the answer. The questioner wanted just the answer.
Burr gets points because he pointed out that there was NO unique answer to the given problem and he offered a simple non-math answer which was ignored.
phoffric gets points because he provided lots of useful links and showed the steps necessary to get (eventually) the correct answer including the difficulty raised by burr.
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