You have the correct answer. It is choice B. Nice work.
K is the midpoint, so [tex]\overline{IK} \cong \overline{GK}[/tex] and [tex]\overline{JK} \cong \overline{HK}[/tex], and along with the congruent vertical angles ([tex]\angle GKH \cong \angle IKJ[/tex] and [tex]\angle GKJ \cong \angle IKH[/tex]), you would use the SAS (side angle side) congruence theorem to prove the inner pairs of triangles to be congruent.
So, [tex]\triangle HKG \cong \triangle JKI[/tex] (top and bottom triangles) and [tex]\triangle GKJ \cong \triangle IKH[/tex] (left and right triangles).
Then through CPCTC, we can show the corresponding pieces are congruent leading to [tex]\overline{GH} \cong \overline{JI}[/tex] and [tex]\overline{GJ} \cong \overline{HI}[/tex] showing the opposite sides of the quadrilateral are congruent. Therefore we do have a parallelogram and enough information to prove it as such.
Side note: CPCTC stands for "corresponding parts of congruent triangles are congruent".
