Why do many hypervalant chemicals have non-constant bond angles?
The $\ce{IF7}$ and $\ce{PCl5}$ molecules have 7 and 5 ligands, and pentagonal and trigonal bipyramidal geometries, respectively. These geometries happen 


  simply because
  there is no geometrical arrangement which can result in five equally
  sized bond angles in three dimensions.


(Wikipedia)

There is no arrangement in which all the angles would be the same, except in a plane, where the repulsion would obviously be a lot higher than in the pyramids.