Questions: What additional information would allow you to prove the quadrilateral is a parallelogram according to the minimum criteria? A) overlineFG overlineEH C) overlineEF congruent to overlineHG

Solution Steps

We are given a quadrilateral EFGH, and we know that FJ = JH and FJ = JH. This means that the diagonal FH is bisected by the other diagonal EG.

We need to find the minimum additional information required to prove that EFGH is a parallelogram. There are several ways to prove a quadrilateral is a parallelogram:

1. Both pairs of opposite sides are parallel.
2. Both pairs of opposite sides are congruent.
3. Both pairs of opposite angles are congruent.
4. One pair of opposite sides is both parallel and congruent.
5. The diagonals bisect each other.

We already know that the diagonals bisect each other. Thus, we need additional information that satisfies one of the other criteria. A) $\overline{FG} \parallel \overline{EH}$ : This states that one pair of opposite sides is parallel. Combined with the bisecting diagonals, this isn't sufficient to prove the quadrilateral is a parallelogram. B) FH || JG. This statement refers to the diagonals, which doesn't help to satisfy any of the minimum criteria. C) $\overline{EF} \cong \overline{HG}$ : This states that one pair of opposite sides is congruent. If we combine this with the fact that the diagonals bisect each other, it is still not enough information to determine that the quadrilateral is a parallelogram. However, if this were combined with option A, it would be enough.

If we are given that one pair of opposite sides are parallel (A), then we need additional information indicating that one pair of opposite sides are congruent or that the other pair of opposite sides is parallel. The given information (C) satisfies this requirement.

Final Answer