To prove the triangles congruent using the SAS (Side-Angle-Side) Congruency Theorem, you need to ensure that you have two sides and the included angle of one triangle congruent to the corresponding two sides and the included angle of the other triangle.
In your responses, you were given:
- \(\angle L \cong \angle W\)
- \(\angle U \cong \angle V\)
- \( \overline{UW} \cong \overline{VL} \)
- \( \overline{UL} \cong \overline{WV} \)
To apply the SAS Congruency Theorem, you should have two sides and the angle formed between those sides congruent in both triangles.
Based on the options provided:
- Options 1 and 2 give angles which do not correspond to the angles formed by the two sides listed in option 3 and option 4.
- Options 3 and 4 give sides, but without knowing if the angle formed between \( \overline{UW} \) and \( \overline{UL} \) is congruent to the angle formed between \( \overline{VL} \) and \( \overline{WV} \), we cannot use SAS.
However, if one of the angle options corresponds to the angle formed by the sides in the other two given options, then we could prove the triangles congruent.
Thus, we would need the measure or congruence of the included angle that is formed by the two sides in one triangle and is congruent to the included angle formed by the corresponding sides in the other triangle.
In conclusion, choose the option that indicates the congruence of the included angle formed by the two sides, which is currently missing from your list. If one of the angles provided is indeed between the corresponding sides, that would be the necessary information to apply the SAS theorem.