Prove vertices of polygon are endpoints of disjoint line segments

If we are given a set of disjoint line segments in the plane, can we prove (or disprove) that we can always join the line segments to make a simple polygon where the vertices of the polygon are the endpoints of the segments and every segment is an edge on the boundary of the polygon.

Here is a simple counter example :

You most join the two leftmost vertices to the two middle vertices, and both ways to do that don't work.

StackExchange Computer Science Q#118332, answer score: 4