Hexagon with Adjustable Angles

This hexagon has images that tessellate by three rotations side to side. But the angles of the side to side rotations are adjustable. The shape won't tessellate for just any angles, but there is at least one set of angles that do work.

Why do those angles work?

How much do they limit the shape of the hexagon?

With the three angles set, how much variety can you generate by dragging the blue vertices?

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Posted at mathhombre.blogspot.com.

John Golden, Created with GeoGebra