The problem asks to identify which quadrilateral could be proved to be a parallelogram by Theorem 6.2D, which states that a quadrilateral is a parallelogram if both pairs of consecutive angles are supplementary (i.e., their sum is $180^\circ$).

Analysis of Quadrilaterals:

1. ABCD: This quadrilateral shows equal opposite sides (parallelogram property), but there's no clear indication of supplementary angles.

2. EFGH: The angles labeled as $180^\circ - x$ and $x$ indicate consecutive angles that sum to $180^\circ$. This satisfies the condition of Theorem 6.2D that consecutive angles must be supplementary.

3. IJKL: It appears to be a rhombus-like shape, but there is no explicit information about consecutive angles being supplementary.

4. MNOP: Diagonals intersect, but there’s no information about angles.

5. QRST: It has some angle measures, but none show consecutive angles summing to $180^\circ$.

Conclusion:

The quadrilateral EFGH is the one that satisfies the conditions of Theorem 6.2D since it shows consecutive angles that are supplementary.

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. What are the properties of a parallelogram?
2. How do you prove a quadrilateral is a parallelogram?
3. What other theorems can be used to prove a quadrilateral is a parallelogram?
4. How does the property of supplementary angles apply to trapezoids?
5. What are the differences between a rhombus and a parallelogram?

Tip: Always check if both pairs of opposite sides and angles meet the parallelogram properties when solving quadrilateral proofs.