Two parallel lines are intersected by a third. Find the angle between the bisectors of the internal same-side angles.

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Use the theorem about the sum of the angles of a triangle.

## Solution

Let lines $l$ and $m$ be parallel, and a third line intersects them at points $A$ and $B$ respectively. Take point $C$ on line $l$ and point $D$ on line $m$ such that these points lie on the same side of line $A B$. Then angles $B A C$ and $A B D$ are consecutive interior angles. By the property of parallel lines, $\angle B A C + \angle A B D = 180^{\circ}$. Let the bisectors of these angles intersect at point $O$. Then $\angle O A B + \angle O B A = 1 / 2 \angle B A C + 1 / 2 \angle A B D = 1 / 2 \cdot 180^{\circ} = 90^{\circ}$. Therefore, triangle $A O B$ is a right triangle.

## Answer

$90^{\circ}$.