The question is asking what additional information is needed to prove that triangles $\triangle ABC \cong \triangle DEF$ using the AAS (Angle-Angle-Side) congruence theorem.

The AAS theorem states that two triangles are congruent if two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle.

From the diagram:

• We already know that $\angle B \cong \angle E$ because both are $30^\circ$, and $\overline{AB} \cong \overline{DE}$ because both sides are labeled as 10 units.
• We need another pair of corresponding congruent angles to use AAS.

Let's analyze the options:

A. $\overline{BC} \cong \overline{EF}$ (This would not prove AAS, because it's a side congruence, not an angle congruence). B. $\angle A \cong \angle C$ (This is the wrong pair of angles; we need corresponding angles). C. $\overline{AC} \cong \overline{DF}$ (This is a side congruence, which does not help with AAS). D. $\angle A \cong \angle D$ (This is the correct pair of corresponding angles. If $\angle A \cong \angle D$, then we would have two angles and a non-included side, satisfying AAS).

Thus, the correct answer is D. $\angle A \cong \angle D$.

Would you like further clarification or details on this? Here are five related questions to deepen your understanding:

1. What is the difference between the AAS and ASA theorems?
2. How do we prove triangle congruence using the SAS theorem?
3. Can the AAS theorem be used if the congruent side is included between the two angles?
4. What are other congruence theorems for triangles besides AAS?
5. In what situations would SSS (Side-Side-Side) be more appropriate than AAS?

Tip: Always make sure to identify the exact parts required for each theorem, such as angles and sides, to avoid mixing them up when proving congruence.