Does each angle of a pentagon measure 108 degrees

If it's a regular pentagon, then yes.

Not necessarily.

In every pentagon, the sum of the 5 inside angles is

(5 - 2) (180) = 540 degrees.

That's always true, no matter what any individual one of the angles is,

or what crazy shape the pentagon has. But the angles don't all need

to be equal.

If the pentagon is a "regular" one, that means that all of its sides

are equal and all of its angles are equal. Then each of the 5 angles

measures 108 degrees.