[ Inequalities with trihedral angles]

In what range can the plane angle of a trihedral angle vary if the other two plane angles are respectively: a) $70^{\circ}$ and $100^{\circ}$; b) $130^{\circ}$ and $150^{\circ}$?
Any dihedral angle of a trihedral angle is less than the sum of the other two dihedral angles; the sum of the dihedral angles of a trihedral angle is less than $360^{\circ}$. Let $\alpha$ be the third dihedral angle of the trihedral angle. Then a)

$$
\alpha<100^{\circ}+70^{\circ}, 100^{\circ}<\alpha+70^{\circ}, 70^{\circ}<\alpha+100^{\circ}
$$

from which we find that $30^{\circ}<\alpha<170^{\circ}$. b)

$$
\begin{gathered}
\alpha<130^{\circ}+150^{\circ}, 130^{\circ}<\alpha+150^{\circ}, \\
150^{\circ}<\alpha+130^{\circ}, \alpha+130^{\circ}+150^{\circ}<360^{\circ},
\end{gathered}
$$

from which we find that $20^{\circ}<\alpha<80^{\circ}$.

## Answer

Y) $30^{\circ}<\alpha<170^{\circ}$; A) $20^{\circ}<\alpha<80^{\circ}$.