Given the right triangular pyramid in the figure, with $\angle APC=\angle APD=\angle ACD =\pi/2$ and $\angle PAC=\phi$, is this enough to completely define the other angles of the pyramid, for example $\angle ADC$? - if so, what is the value of $\angle ADC$?
Line $AP$ is perpendicular to plane $PDC$ and $AC$ is perpendicular to $CD$. By the three perpendicular theorem it follows that line $CD$ is perpendicular to $PC$.
Hence point $D$ can be taken everywhere, on the line of plane $PCD$ passing through $C$ and perpendicular to $PC$, without changing the given four angles. But if $D$ changes then $\angle ADC$ changes too. The given data are thus not enough to completely define the other angles of the pyramid.