[![triangle face of tetrahedron][1]][2]

As [Li Li commented on your closely related Question](https://math.stackexchange.com/questions/2249914/solving-for-angles-in-a-tetrahedron#comment4626968_2249914), the [Law of Cosines](https://en.wikipedia.org/wiki/Law_of_cosines) gives us a system of three quadratic equations for unknowns $x_1,x_2,x_3$:

$$ \begin{align*} 
x_1^2 + x_2^2 - d_3^2 &= 2x_1 x_2 \cos \theta_3 \\
x_1^2 + x_3^2 - d_2^2 &= 2x_1 x_3 \cos \theta_2 \\
x_2^2 + x_3^2 - d_1^2 &= 2x_2 x_3 \cos \theta_1 
\end{align*} $$


 [1]: https://i.sstatic.net/LOzNhm.png
 [2]: https://i.sstatic.net/LOzNh.png