Sydler pi/4 polyhedron: Step 4
Consider a polyhedron with 6 triangles and label one pair of opposite edges by 2 and 3. We can find a such a polyhedron Q (green) such that the angle at the edge labeled 2 is π - α and all unlabeled edges have right angles. We call the angle at edge 3 β.
If we remove Q from one place and add it along the edge with angle α, we achieve that the two extra angles (labeled 3 and 4) of the resulting polyhedron are now along edges parallel as line segments (in the sense that they span a rectangle and not just a parallelogram).