The problem asks you to determine what additional information is required to know if two triangles are congruent using the HL (Hypotenuse-Leg) theorem.

Here are the key features of the problem:

1. Given Triangles: The diagram shows two right-angled triangles, $\triangle WVX$ and $\triangle ZYX$, with shared segment $XZ$, which seems to represent the right angle.

2. HL Theorem: The Hypotenuse-Leg theorem applies to right-angled triangles. It states that two triangles are congruent if their hypotenuses and one corresponding leg are congruent.

To apply the HL theorem, we need to know:

• That both triangles are right-angled.
• The hypotenuses of the two triangles are congruent.
• One leg from each triangle is congruent.

In this case, since both triangles appear to be right-angled, and they share one leg $XZ$, we need to confirm the congruence of the hypotenuses.

Answer:

• Option 1: $\overline{VW} \cong \overline{ZY}$ is the correct additional information.
  • These represent the hypotenuses of the two triangles, and proving them congruent would satisfy the Hypotenuse-Leg criterion for triangle congruence.

Let me know if you'd like more details!

Here are 5 questions to extend this problem:

1. What is the Hypotenuse-Leg (HL) theorem and how is it used in triangle congruence?
2. How can you determine congruence of right triangles using the hypotenuse and a leg?
3. What are the other triangle congruence theorems besides HL?
4. How does sharing a leg in a triangle impact the congruence proof?
5. Why is proving the hypotenuse congruent necessary in the HL theorem?

Tip: When working with right triangles, the HL theorem is often quicker than other congruence theorems!