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Dealing with oblique projections, I encountered a problem that I can formulate like this:

I have a parallelogram with two sides of 1 unit and two unknown. I also know the angles. There is a right triangle sharing the parallelograms unknown side, like in the image. Is it possible to figure out the lengths of the triangle, especially x? enter image description here

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  • $\begingroup$ That is not a possible triangle, the hypotenuse (with length $2$) can't be shorter than one of its legs (with length $4$?) $\endgroup$ – user170231 Mar 25 at 18:45
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    $\begingroup$ @user170231 those are letters, not numbers. $\endgroup$ – Paul Mar 25 at 18:49
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    $\begingroup$ @user170321 I'm sorry for my writing, the "2" is a z and the "4" is y $\endgroup$ – Andrei Agache Mar 25 at 18:49
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If I am reading your handwriting correctly, you have 1 angle and 1 side length (though it is a parallelogram and that gives you the opposite side length and all 4 angles). In this case, just imagine lengthing the side labeled $z$ by a lot (think 100x as long), then $x$ would be made much longer, without changing the length of the side labeled 1 or any angles. Thus $x$ cannot be found with only 1 angle and 1 side length from this parallelogram.

Though you could solve for $x$ in terms of $z$.

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  • $\begingroup$ I am unfamiliar with geometry. I know with Pythagoras theorem you can solve for $x$ in terms of $y$ and $z$. I know with trig functions, I can solve for $x$ in terms of $z$. What is another way to solve for $x$ in terms of $z$? $\endgroup$ – name Mar 25 at 19:29
  • $\begingroup$ the angle right next to the 53 degree one, must add with the 53 degree one to ninety, so it is a 37 degree angle. The relation is via a trig function, so you do need trig to solve for x in terms of z. Using Trig, sin(53 degrees) = x/z , so x= zsin(53 degrees). That is the sine of an angle is equal to the opposite side divided by the hypotenuse (this can be taken as a definition of the sine function) $\endgroup$ – Mark Mar 25 at 21:20
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Yes, you could find $x$ and $y$ in terms of $z,$ but you don't know what $z$ is.

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