[![enter image description here][1]][1]

The figure shows a hexagon and triangle tiled by six tiles, which are pairwise congruent.

My question is:

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

Small print: 

 - I define a polygonal tile as a figure comprised of:

     A finite set $P$ of at least three points.

     The (straight) lines $X\pi(X)$ for each point $X\in P$ and some cyclic permutation $\pi$ of $P$ such that distinct lines do not intersect except at mutual end points. 

     The region contained by the lines.

 - The hexagon, triangle and tiles are assumed to include the vertices, edges and interior.

 - The figures tiled must be the union of exactly one congruent copy of each tile (though the set of tiles itself is allowed to contain congruent figures). No copy of a tile can share an interior point with the copy of any other tile, but may share boundary points with one or more.

 - A congruent copy of a tile may be a "flipped" version (e.g. the cyan tiles in the above diagram are actually mirror images of each other - this is allowed).  

  [1]: https://i.stack.imgur.com/fxjW6.png