Spoiler for Assuming that I understood the definition of "bent" correctly
I'm thinking of how to fill up the entire angle A in k-man's drawing.
The reflex interior angle would have to approach 180 degrees to do that.
That demands an infinite number of bents, whose angles approach 180 degrees as a limit.
The requirement of an infinite number of bents contradicts the terms of the OP.
Q. E. D.