It is easy to see that for isosceles tetrahedron (https://en.wikipedia.org/wiki/Disphenoid) all faces are equal acute triangles. If we consider regular tetrahedra and attach a regular triangular pyramid to each face, then we obtain a polyhedra with equal triangular faces; each face will be an isosceles triangle with obtuse angle $< 120^{\circ}$.

For which other triangles there is a polyhedra with all faces equal to this triangle?

The same question for quadrangles (see https://en.wikipedia.org/wiki/Trapezohedron for examples).