Suppose L and L' are parallel.

Let P denote the intersection of L with AB and let Q denote the intersection of L' with BC. Then PQ forms a transversal that cuts across both lines L and L'. Then since consecutive interior angles are supplementary, we can conclude that angles PQB and QPB are supplementary. Further, since the angles in a triangle add to 180 degrees, we can conclude that

angle ABC = angle PBQ

has angle measure 0.

This implies AB and BC are parallel.