## Equal angles in tetrahedron

If the angles between the three pairs of opposite sides of a tetrahedron are equal, prove that these angles are right.

**Solution** We use the identity . Since the angles between opposite sides is the same for all three pairs of opposite edges, let’s say , we can conclude from the inner product properties that

.

Now suppose the second factor is 0. Then we can’t have all signs to be + or all to be – and we will get a formula like or something equivalent with this. So we have equality in Ptolemeu’s inequality, which means that are in the same plane and on the same circle, which is a contradiction with the fact that is a tetrahedron.

By the arguments above, the second factor of our product can never be 0, meaning that , which means that all the angles between opposite sides are right ( )