Problem 3. One of the acute angles of a right triangle is $35^{\circ}$. What is the angle formed by the bisector of the largest exterior angle and the line to which the shortest side of the triangle belongs?
Solution. Let the right angle in $\triangle A B C$ be at vertex $C$ (see diagram). The acute angles of the triangle are $\alpha=55^{\circ}$ and $\beta=35^{\circ}$. The exterior angles are supplementary to the corresponding interior angles of the triangle and are equal to $125^{\circ}, 145^{\circ}$, and $90^{\circ}$. The smallest side lies opposite the smallest interior angle, which is side $A C$. The bisector of the largest exterior angle $\beta_{1}$ intersects the line $A C$ at point $D$. The required angle is the acute angle of the right triangle $B C D$, so we sequentially obtain

![](https://cdn.mathpix.com/cropped/2024_06_05_b353751bdd87cc544500g-08.jpg?height=435&width=1007&top_left_y=429&top_left_x=381)

Thus, the required angle is equal to $17^{\circ} 30^{\prime}$.