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Another dissection puzzle

The figure shows a hexagon and triangle tiled by six identical tiles.

My question is:

What is the smallest number of polygonal tiles that will tile both a regular hexagon and an equilateral triangle in such away that all edges of the tiles are all parallel to an edge of the tiled figure?

Small print:

  • I define a polygonal as a figure comprised of a finite set P of at least three points, together the lines x permuation $\pi$