I don't think the figures you listed can be put in any one "correct" order. There are multiple ways to make a tree, depending on the properties you consider first. You could say that the square is a kind of rhombus, and the rhombus is a kind of kite (if you use an inclusive definition); and if you are defining trapezoid in the British sense of having no parallel sides, you can say that the kite is a kind of trapezoid. But that is not the order you say they gave.

If we focus on symmetry, the square has four lines of symmetry, the rhombus two, the kite one, and the trapezoid none, so this results in the same order.

I am wondering if it is really intended to be just a discussion-starter, rather than a problem with one correct answer. I couldn't translate the PDF you provided, but in part it looked like a discussion of teaching, which may have been focused on how to discuss such questions.

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