Q:

Why do many hypervalant chemicals have non-constant bond angles?

According to the VSEPR model, when a molecule in formed in the shape of $\ce{AB_{x}}$,

like $\ce{IF7}$, $\ce{SCl6}$, and $\ce{PCl5}$,

its structure is found by reducing the repulsions between peripheral atoms. This gives rise to many shapes, most of which have constant bond angles. Examples of those with constant bond angles include

$\ce{CH4}$ with 109.5° bond angles
$\ce{CO2}$ with 180° bond angles.

My question is then, why do some have non-constant bond angles?
The following do not

$\ce{IF7}$ with 72° and 90° bond angles
$\ce{PCl5}$ with 120° and 90° bond angles.

A:

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.

