Can these "Simple Proofs of a Rectangle Tiling Theorem" be applied to the tiling with triangles?

Question

In this Cambridge paper there are proofs for these three theorems:

1. If a finite number of rectangles, every one of which has at least one integer side, perfectly tile a big rectangle, then the big rectangle also has at least one integer side.
2. If a finite number of rectangles, every one of which has at least one rational side, perfectly tile a big rectangle, then the big rectangle also has at least one rational side.
3. If a finite number of rectangles, every one of which has at least one algebraic side, perfectly tile a big rectangle, then the big rectangle also has at least one algebraic side.

While I can follow these proofs somewhat well I don't really see how generally they can be applied. So could you prove the same way that

"If a finite number of triangles, every one of which has at least one even integer height, perfectly tile a big rectangle, then the big rectangle also has at least one even integer side."

?

I think that this should be possible but I can't transfer the idea of the paper onto a new/ differen concept.
I suppose one could proceed by showing that a triangle having one even height is similar to a rectangle having one even integer side and then you could maybe transport the idea of the paper...

Answer 1

The conjecture as stated is false. Consider a $\sqrt{2}\times \sqrt{2}$ square cut into $4$ right triangles as shown:

Each of the triangles has two integer heights of $1$, but the rectangle has no integer (or even rational) dimensions. (If you specifically want the heights to be even integers, just scale everything up by $2$.)