Given a triangle with one obtuse angle, is it possible to cut the triangle into smaller triangles, all of them acute? (An acute triangle is a triangle with three acute angles. A right angle is of course neither acute nor obtuse.) If this cannot be done, give a proof of impossibility. If it can be done, what is the smallest number of acute triangles into which any obtuse triangle can be dissected?

The illustration shows a typical attempt that leads nowhere. The triangle has been divided into three acute triangles, but the fourth is obtuse, so nothing has been gained by the preceding cuts.

fix tex

It isn't possible. In fact, no triangle can be cut into smaller triangles, all of them acute.

1. At some point or another, you will have to bisect the obtuse angle in your initial triangle. Otherwise it will stay obtuse.
2. By cutting a triangle into two smaller triangle by drawing a line from one corner to the opposite vertex, at least one of the resulting triangles will be right or obtuse. (because 180°≥2×90°)
2.1. There is another way of cutting a triangle into smaller triangles, which is to draw three lines between each corner and a central point. In this case too, at least one of the resulting triangles will be obtuse, because 360°>3×90°.

P76283
It is possible.