Inclined parallelepiped volume

Let there be a parallelepiped ABCDA1B1C1D1. Draw a plane perpendicular to the base ABCD through the edge BC and add the inclined parallelepiped with a triangular prism BB1B2CC1C2. We now cut off the triangular prism from the resulting body with a plane passing through the edge AD and perpendicular to the base ABCD. Then we get the parallelepiped again. This parallelepiped has a volume equal to the volume of the original parallelepiped.

The completed prism and the cut-off are combined by parallel transfer to the segment AB, therefore, they have the same volumes. In the described transformation of the parallelepiped, the area of ​​its base and height are preserved. The planes of the two side faces are also preserved, while the other two become perpendicular to the base.

Applying such a transformation once again to inclined faces, we obtain a parallelepiped, in which all the lateral faces are perpendicular to the base, i.e. straight parallelepiped.