[ $[$ straight lines and planes in space]

Points $M$ and $N$ lie on the edges $B C$ and $A A 1$ of the parallelepiped $A B C D A 1 B 1 C 1 D 1$. Construct the point of intersection of the line $M N$ with the plane of the base $A 1 B 1 C 1 D 1$.

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Let's draw an auxiliary plane through the line $A A 1$ and point $M$. This plane has a common point $M$ with the plane of the face $B B 1 C 1 C$ and passes through the line $A A 1$, which is parallel to the plane of the face $B B 1 C 1 C$ (since $A A 1 \| B B 1$). Therefore, the line of intersection of these planes is parallel to $A A 1$. Let $M 1$ be the point of intersection of this line with the edge $B 1 C 1$, and $P$ be the point of intersection of the lines $M N$ and $M 1 A 1$, which lie in the auxiliary plane. Since the line $M 1 A 1$ lies in the plane of the base $A 1 B 1 C 1 D 1$, $P$ is the desired point of intersection of the line $M N$ with the plane of this base.