  We say that a triangle $T$ tiles a polygon $A$, if $A$ can be dissected into
finitely many nonoverlapping triangles similar to $T$. We show that if $N>42$,
then there are at most three nonsimilar triangles $T$ such that the angles of
$T$ are rational multiples of $\pi$ and $T$ tiles the regular $N$-gon.
  A tiling into similar triangles is called regular, if the pieces have two
angles, $\al$ and $\be$, such that at each vertex of the tiling the number of
angles $\al$ is the same as that of $\be$. Otherwise the tiling is irregular.
It is known that for every regular polygon $A$ there are infinitely many
triangles that tile $A$ regularly. We show that if $N>10$, then a triangle $T$
tiles the regular $N$-gon irregularly only if the angles of $T$ are rational
multiples of $\pi$. Therefore, the numbers of triangles tiling the regular
$N$-gon irregularly is at most three for every $N>42$.
