Answer: 30. Solution: Estimation. There are 6 pairs of points, that is, 6 possible lines. According to the condition, there are no parallel pairs among them. Draw six lines parallel to them through one point. They will divide the plane into 12 angles, so there is an angle no more than 30 degrees. Example. Take an arbitrary segment $A C$ and construct an equilateral triangle $A B C$ and an isosceles triangle $A D B$ with an angle of $120^{\circ}$ at vertex $D$ on opposite sides of it. It is easy to verify that the angle between any two of the six lines defined by the points $A, B, C, D$ is no less than 30 degrees.
5. Petya marks four points on a plane so that all of them cannot be crossed out by two parallel lines. From the lines passing through pairs of points, Vasya chooses two, measures the angle between them, and pays Petya an amount equal to the degree measure of the angle. What is the largest sum Petya can guarantee for himself?