Problem 3. Let $\alpha$ and $\beta$ be two adjacent angles such that their bisectors are mutually perpendicular. If it is known that $\alpha$ is $30^{\circ}$ greater than $\beta$, calculate how many degrees each of them measures.
Solution. Since the angles are adjacent and their bisectors are mutually normal, we get $\frac{\alpha}{2}+\frac{\beta}{2}=90^{\circ}$, which means $\alpha+\beta=180^{\circ}$. Since we know that $\alpha$ is $30^{\circ}$ greater than $\beta$, we get $\alpha=\beta+30^{\circ}$. If we substitute this into $\alpha+\beta=180^{\circ}$, we get $\beta=75^{\circ}$. Therefore, $\alpha=105^{\circ}$.