How does knowing the properties and definitions of two - dimensional figures helps you classify them in a hierarchy diagram?

Answer 1

I am not sure that it does, because there is no clear hierarchy.

For example, with quadrilaterals (moving towards regularity), you could go

parallelogram -> rectangle -> square

or

parallelogram -> rhombus -> square

In the first case, the number of equal angles has been increased first and then the number of sides of equal length. In the second case it is the other way around.

But neither of

parallelogram -> rectangle -> rhombus -> square

nor

parallelogram -> rhombus -> rectangle -> square

is a valid hierarchy.

So there is no hierarchical order between a rectangle and rhombus. The same situation applies to all polygons with more sides.