Calculate the angles It is given that that $AB \parallel CD,\ AC \parallel DE$ and $BD \parallel CE$ and the question is to give the value of the angles $x$ and $y$ if we know the value of the angles as shown in the picture. It is also known that $x,y \in \{20,30\}$.

 A: There is no solution to this problem. You can consider the following coordinates:
\begin{align*}
A &= \lambda\begin{pmatrix}-\cot20°\\1\end{pmatrix} &
B &= \lambda\begin{pmatrix}\cot30°\\1\end{pmatrix} \\
C &= \begin{pmatrix}\cot20°\\-1\end{pmatrix} &
D &= \begin{pmatrix}-\cot30°\\-1\end{pmatrix} \\
\end{align*}
You actually don't need $E$ at all. Now as you vary $\lambda$, the points $A$ and $B$ will move on their respective rays $CA$ resp. $DB$ arbitrarily. You have
\begin{align*}
\tan(x+30°)&=\frac{1+\lambda}{\cot30°-\lambda\cot20°} &
\tan(y+20°)&=\frac{1+\lambda}{\cot20°-\lambda\cot30°}
\end{align*}
So up front, neither angle is fixed. But if you assume $x,y\in\{20°,30°\}$ then you get contradictions:
$$\begin{array}{ccc}
x&y&\lambda\\\hline
20°&\phantom08.334°&0.249\\
30°&12.122°&0.347\\
50°&20°\phantom{.000}&0.532\\
70°&30°\phantom{.000}&0.742
\end{array}$$
So as you see, you can't choose $x$ and $y$ from the set $\{20°,30°\}$ at the same time. You can't even choose them from the interval $[20°,30°]$ at the same time.

