The three parts are faces, vertices and edges. The faces are the flat surfaces, the edges are the lines that joint the corners which are called vertices.
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They don't have names beyond "the faces of the tetrahedron."
Tetrahedron.
A tetrahedron or triangular pyramid.
They do not have specific names. They are triangles and, if one of them happens to be horizontal and lower than the rest, it is called the base.
Well, a tetrahedron has four sides. A regular tetrahedron is a triangular pyramid.