This puzzle appeared in Microsoft Puzzle Hunt 19, whose theme was TV.
That event was modular in that each author wrote a module consisting of a few
puzzles and a metapuzzle.
The theme of my module was Saturday Night Live.
The real word NONAGON means "nine-sided figure", and the prefix "NON" as used in the pre-taped answer "NONBELIEVER" means "opposite of". NONAGON, of course, isn't using NON in that sense. But, if it did, then perhaps the word AGON would mean the opposite of a nonagon. Thus, AGON is the answer to the clue "geometric figure having eight or fewer, or ten or more, sides".