For a certain triangle, the ratio of two exterior angles is $\frac{13}{9}$; the difference between these angles is $45^{\circ}$. What are the angles of the triangle?
$$
\alpha=33^{\circ} 45^{\prime}, \quad \beta=78^{\circ} 45^{\prime} \quad \text { and } \quad \gamma=67^{\circ} 30^{\prime} \text {. }
$$

Number of solutions: 25.