My question in this post is, why do signs with large angles fall? For example, fold a piece of paper in half, set the angle between the two halves, and place it on the floor; it will stand or collapse depending on that angle. I currently believe that the only horizontal force affecting the sign is friction; which does not vary with angle, as friction is $F_n \cdot F_s$ and both are not related to the angle. Any help on this would be appreciated.
It is due to tensile internal forces (and the center of mass doesnt change position). And you are correct, the friction doesnt change. Look at the graph for the forces acting on the small element of paper in contact with the floor. We have two equations to describe vertical and horizontal forces at equilibrium:
From which you get
if $\alpha$ is too large, the equality no longer holds and the horizontal tension overcomes the friction.
I believe a major factor in it is failure of the central pivot's friction to resist the larger torque generated by the longer lever arms.
So, yes, this is a matter of friction, but not only on the feet. What resists "first" is the friction of the central pivot. If that fails, then feet friction comes into play.