Problem 2. The bisectors of two adjacent angles $\alpha$ and $\beta$ intersect at a right angle. Calculate the angles $\alpha$ and $\beta$ if $\alpha-\beta=30^{\circ}$.
Solution. For the adjacent angles $\alpha$ and $\beta$, from the condition for the angle bisectors, we have $\frac{\alpha}{2}+\frac{\beta}{2}=90^{\circ}$, which means $\alpha+\beta=180^{\circ}$, i.e., the required angles are supplementary. Then, from $\alpha-\beta=30^{\circ}$ we have $\beta=\alpha-30^{\circ}$ and $\alpha+\left(\alpha-30^{\circ}\right)=180^{\circ}$, from which we get $2 \alpha=210^{\circ}$, i.e., $\alpha=105^{\circ}$. Then $\beta=75^{\circ}$.