To prove that triangles \( \triangle ABC \cong \triangle QRS \) using the Angle-Side-Angle (ASA) criterion, you need two angles and the included side (the side between the two angles) of one triangle to be congruent to the corresponding two angles and the included side of the other triangle.
Given the options, you would need either:
- \( \angle A \cong \angle Q \) — this indicates that one pair of corresponding angles is congruent.
- \( \angle C \cong \angle S \) — this indicates that another pair of corresponding angles is congruent.
- \( \overline{AC} \cong \overline{QS} \) — this indicates that the included sides (the side between the angles) are congruent.
Out of these, \( \overline{AC} \cong \overline{QS} \) is the most crucial piece of information needed to apply the ASA criterion. With \( \angle A \cong \angle Q \) and \( \angle C \cong \angle S \) (which are angles), you ultimately need the congruence of the included sides \( \overline{AC} \cong \overline{QS} \) to complete the ASA condition.
Therefore, the correct response is:
\( \overline{AC} \cong \overline{QS} \)
This is necessary to establish the congruence of the two triangles using the ASA criterion.