Final Answer:
Additional information that could be used to prove △EFG ≅ △E'F'G' using AAS is eg = 12 and e'g' = 12.
Step-by-step explanation:
In the given triangles △EFG and △E'F'G', the angles ∠FEG and ∠F'E'G' are given as 72°, while ∠EGF and ∠E'G'F' are 66°. To establish congruence using the AAS criterion, we require one more pair of corresponding equal angles. Selecting the information eg = 12 and e'g' = 12 becomes crucial for this proof.
By employing the Side-Angle-Side (SAS) congruence criterion, we can now show that △EFG ≅ △E'F'G'. The angles ∠FEG and ∠F'E'G' serve as the angle components, and the sides EG and E'G', being equal in length (eg = e'g' = 12), constitute the side component. This fulfills the conditions for the AAS congruence, as both triangles share an equal side and a pair of corresponding equal angles.
Other provided options do not offer a complete set of conditions for proving AAS congruence. Equal side lengths alone, as in fg = 15 and f'g' = 15, are insufficient without a corresponding pair of equal angles. Likewise, having equal angles or approximately equal sides does not satisfy the AAS congruence criteria. Therefore, the specified information, eg = 12 and e'g' = 12, uniquely satisfies the AAS conditions and allows us to conclude that △EFG ≅ △E'F'G'.