Question

Please help

"Is there enough information to prove quadrilateral GHIJ
is a parallelogram?"
Select the answer that is completely correct.

Answer 1

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".