The problem you're working on involves proving triangle congruence using the Angle-Side-Angle (ASA) criterion, which states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.

From the image, it appears that additional information is needed to confirm the congruence between two triangles. The question asks what additional information is required to make the triangles congruent by ASA.

To apply ASA, you already need:

1. Two pairs of corresponding angles that are congruent.
2. The side between these two angles (the included side) must also be congruent.

Analysis of the options:

1. LJ ≅ WU (Corresponding sides are congruent) — This could work if it's the included side between two angles.
2. VJ ≅ VW (Different sides congruent) — Not relevant unless it's the included side.
3. ∠W ≅ ∠J (Angles congruent) — This would confirm the second pair of congruent angles.
4. ∠U ≅ ∠L (Another pair of angles congruent) — Also possible but depends on the configuration.

Since the question focuses on ASA, we need to ensure that two pairs of angles are congruent along with the included side. Therefore, the correct additional information is most likely ∠W ≅ ∠J (the third option), as this would provide two pairs of angles and the congruent included side necessary for ASA congruence.

Would you like a detailed explanation of ASA or have any questions?

Related questions:

1. What is the ASA (Angle-Side-Angle) triangle congruence criterion?
2. How is the ASA rule different from other congruence rules like SSS or SAS?
3. What role do corresponding sides and angles play in triangle congruence?
4. Can two triangles be congruent if only angles are congruent (without any side congruence)?
5. What are some practical applications of triangle congruence in real-world scenarios?

Tip:

Always verify that the side in question is the included side between two congruent angles when applying the ASA rule.