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The analysis is wrong. There is no reason to believe that AD and BE intersect, etc.

If AD and BE don't intersect then the quadrilateral ADBE doesn't lie in a plane, and instead you have a curved surface there, which is not what you'd expect from the diagram. The idea is that if you assume all of these surfaces are flat, nothing lines up and the shape can't exist. Certainly you can make something that looks like this, but it won't have perfectly flat surfaces everywhere.

It states that "The drawing appears to represent a polyhedron with two triangular faces and three quadrilateral faces."

There is no reason to believe it actually is a quadrilateral. The initial question just gives you a drawing...

It is not a pyramid either, since all the lines would intersect at the apex.

So what you have is a pyramid-like drawing and some wrongful analysis of it ¯\_(ツ)_/¯

The only other interpretation is that this is just a rough sketch and actually G=H=I and the analysis again is incorrect, since GHI isn't a triangle, it's the same point, the apex, which thus can exist in both planes.

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