Can you prove def is congruent to hgf? justify your answer

You are given that DE is congruent to GH (both have two hash marks). Since these are opposite angles EFD and HFG these angles are congruent (corresponding angles opposite congruent sides of a triangle are congruent). It is also true that these angle are vertical angles so you could prove they are congruent that way as well.

DF is congruent to FG (they both have 1 hash mark) so the angles opposite them are congruent (DEF and HGF). 

Now you have two angles in one triangle congruent to two angles in the pother. Since the angles in any triangle sum to 180, if two are congruent then the third must be as well so we know that angle D and angle G are congruent.

This is all we need to prove the triangles are congruent by SAS congruent to SAS. Specifically S (DE and GH), A (D and G) and S(DF and GF).