Why does not a tetraheron have one pair of parallel faces?

Answer 1

Each face of a polyhedron is a polygon which has at least three straight sides; all the polygons of the polyhedron join along one of their sides to another polygon to form the edges of the polyhedron.

Now consider when you have two parallel faces of a polyhedron:

1. They do not meet (as they are parallel) and so have no side in common to form an edge of the polyhedron;
2. There must be at least three sides to each polygon which means there must be at least three more polygons to join the two faces together (one for each side of the parallel faces);
3. Thus there are at least 2 + 3 = 5 faces to the polyhedron, ie the polyhedron has 5 or more faces.

A tetrahedron has 4 faces which is less than 5 faces.

Thus two of the faces of a tetrahedron cannot be parallel.

The smallest polyhedron which can have a pair of parallel faces is a pentahedron with two triangular parallel faces and three quadrilateral faces joining them (for example a tetrahedron with one corner cut off by a plane parallel to the opposite face).