"World's Hardest Easy Geometry Problem"

Question

This question is a "corollary" (if you will) to the World's Hardest Easy Geometry Problem (external website). Formally, this is called Langley's Problem. The objective of that problem was to solve for angle $x^{\circ}$, with the given angles of $10^{\circ}, 70^{\circ}, 60^{\circ}, 20^{\circ}$. Someone presented a solution to that problem. Here's also a rather colorful and interactive solution to a problem like this, but with different angles.

Now, I wanted to generalize this problem, replacing the angles of $10^{\circ}, 70^{\circ}, 60^{\circ}, 20^{\circ}$ with angles of $W^{\circ}, X^{\circ}, Y^{\circ}, Z^{\circ}$, respectively (see below picture).

How can we derive an analytical expression of angle $x^{\circ}$, in terms of $W^{\circ}, X^{\circ}, Y^{\circ}, Z^{\circ}$?

Answer 1

There is in principle no problem in obtaining an expression for the top angle in terms of the bottom ones. Let the unlabelled intersection point in the picture be $I$. Let the bottom side be $1$. We can use the Sine Law to find $AI$ and $BI$. We can also find $AD$, and $BE$, and now we know $ID$ and $IE$, so we can solve for the mystery angle.

This does not result in a nice expression for the mystery angle, but it is an expression.

Answer 2

Well, I'll give a guide to follow, not a final expression. I called you unknown angle as $\alpha_9$ (in order not to mess, because you have a big $X$ and a small $x$).
Since you know $x$ and $y$ then you know $\alpha_4$.
Since you know $x$, $w$, $z$ and $y$ then you know $\alpha_1$.
Since you know $x$ and $\alpha_5$ then you know $\alpha_3$.
Since you know $w$ and $\alpha_5$ then you know $\alpha_6$.
Then you end up with a system of $4$ equations: $$ \begin{cases} \alpha_1+\alpha_2+\alpha_7=\pi \\ \alpha_4+\alpha_8+\alpha_9=\pi \\ \alpha_2+\alpha_3+\alpha_9=\pi\\ \alpha_6+\alpha_7+\alpha_8=\pi \end{cases} $$ with $4$ unknowns: $\alpha_2, \alpha_7, \alpha_8, \alpha_9$. And $\alpha_9$ is what you are looking for.

Answer 3

I have found the equation!

I first tried to insert the latex equation directly here, but that was ugly so I rendered it separately and inserted an image instead:

Answer 4

Refering to the Triangle Problem 1 diagram: If z>=w then x=80-w+arctan[(tan(z+10)-tan(w+10))/(tan(10)^2-tan(z+10)*tan(w+10))] degrees. There is no general solution for x in terms of z and w without using trigonometry since some solutions for x require an infinite series of digits to the right of the decimal point.