
The minimal number of acute triangles is 7. The
pattern for the seven triangles is shown in the illustration. An
elegant proof for the seven acute triangles has been provided by
Wallace Manheimer in American Mathematical Monthly, November 1960. The
logic behind the proof is as follows.
The obtuse angle must be divided by a line. This line cannot go all
the way to the other side, for then it would form another obtuse
triangle (or two triangles with right angles), which in turn would
have to be dissected, consequently the pattern for the large triangle
would not be minimal. The line dividing the obtuse angle must,
therefore, terminate at a point inside the triangle. At this vertex,
at least five lines must meet, otherwise the angles at this vertex
would not all be acute. This creates the inner pentagon of five
triangles, making a total of seven triangles as shown in the
illustration. 
