Question

Please help

Answer 1

The correct answer is B!!!

Answer 2

You have the correct answer. It is choice B. Nice work.

K is the midpoint, so $\overline{IK} \cong \overline{GK}$ and $\overline{JK} \cong \overline{HK}$, and along with the congruent vertical angles ($\angle GKH \cong \angle IKJ$ and $\angle GKJ \cong \angle IKH$), you would use the SAS (side angle side) congruence theorem to prove the inner pairs of triangles to be congruent.

So, $\triangle HKG \cong \triangle JKI$ (top and bottom triangles) and $\triangle GKJ \cong \triangle IKH$ (left and right triangles).

Then through CPCTC, we can show the corresponding pieces are congruent leading to $\overline{GH} \cong \overline{JI}$ and $\overline{GJ} \cong \overline{HI}$ 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".