On face angles of tetrahedra with a given base

Let us consider the set $\mathrm{\Omega }(\mathrm{△}ABC)$ of all tetrahedra $ABCD$ with a given non-degenerate base $ABC$ in ${\mathbb{E}}^3$ and $D$ lying outside the plane $ABC$. Let us denote by $\mathrm{\Sigma }(\mathrm{△}ABC)$ the set $\left\{\right(\cos \bar{\alpha },\cos \bar{\beta },\cos \bar{\gamma })\in {\mathbb{R}}^3\mathrm{\mid }ABCD\in \mathrm{\Omega }(\mathrm{△}ABC)\}$, where $\bar{\alpha }=\mathrm{\angle }BDC$, $\bar{\beta }=\mathrm{\angle }ADC$, and $\bar{\gamma }=\mathrm{\angle }ADB$. The paper is devoted to the problem of determining of the closure of $\mathrm{\Sigma }(\mathrm{△}ABC)$ in ${\mathbb{R}}^3$ and its boundary.