Yes its a postulate but what makes the other pair parallel
I encountered a proof problem where I was given parallelogram ABCD and that AE is congruent to FC. I had to prove EBFD was a parallelogram. I want to use the theorem that if one pair of sides of a quadrilateral is both congruent and parallel, then it is a parallelogram. In doing so, I would like to say that EB is parallel to FD because parts of parallel lines are parallel. Is that a postulate?
Right, I understand that I can do it with angles but what was first apparent to me was using the theorem that if a quadrilateral has one pair of opposite sides that is both parallel and congruent then it is a parallelogram. Anyway, I wrote this:
S: (line segment)AB || (line segment)DC R: Opposite sides of parallelograms are parallel.
S: (line segment)EB || (line segment)DF R: Line segments contained within parallel lines are parallel.
I wasn't sure if somewhere in the proof I should be including another statement that says (line)AB || (line)DC
"Parts of parallel lines are parallel" is neither a postulate nor a theorem because we do not define "parallel segments". I think what you want to say is not that "part of the line" or "segments of the line" are parallel but that the two segments lie on parallel lines. Once you have said that you can use the fact that the lines, not the segments, are parallel.