Try this:
FA/AB = AE/AD by the side-splitter theorem because EF ║ BD
∠FAE ≅ ∠BAD because an angle is congruent to itself
AE/AD = EG/DC by the side-splitter theorem because EG ║ CD
∠AEG ≅ ∠ADC because they are corresponding angles where a transversal crosses parallel lines
quadrilateral AFGE is similar to quadrilateral ABCD by SASAS similarity of quadrilaterals
From here, there are numerous ways to reach the conclusion. One of them is ...
∠AFG ≅ ∠ABC by corresponding angles of similar quadrilaterals
∴ FG ║ BC because ∠AFG and ∠ABC are corresponding angles where transversal AB crosses FG and BC
