$\left.\begin{array}{l}{\left[\begin{array}{l}\text { Ratio of volumes }\end{array}\right]} \\ {[\text { Volume of tetrahedron and pyramid }]}\end{array}\right]$

The volume of the tetrahedron $A B C D$ is $V$. Points $M$ and $N$ are taken on edge $A B$, and points $P$ and $Q$ are taken on edge $C D$. It is known that $M N=\alpha A B, P Q=\beta C D$. Find the volume of the tetrahedron $M N P Q$.
Let $d$ be the distance between the lines $AB$ and $CD$, and $\phi$ be the angle between them. Then,

$$
V=\frac{1}{6} AB \cdot CD \cdot d \sin \phi
$$

Therefore,

![](https://cdn.mathpix.com/cropped/2024_05_06_1f91526468b1b4d32730g-15.jpg?height=97&width=546&top_left_y=555&top_left_x=756)

![](https://cdn.mathpix.com/cropped/2024_05_06_1f91526468b1b4d32730g-15.jpg?height=89&width=546&top_left_y=727&top_left_x=756)

## Answer

$\alpha \beta V$.