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The problem provided involves using the **Hypotenuse-Leg (HL) Congruence Theorem** to prove that two right triangles, \( \triangle ABC \) and \( \triangle XYZ \), are congruent. 

### Given:
- \( \angle B \cong \angle Y \)
- Hypotenuses \( \overline{AC} \) and \( \overline{XZ} \) are congruent.

### HL Congruence Theorem:
The HL Congruence Theorem states that **two right triangles are congruent if their hypotenuses and one leg are congruent**.

### What We Need:
To apply the HL Congruence Theorem, we already know:
- The hypotenuses \( \overline{AC} \cong \overline{XZ} \).
- The right angles \( \angle B \cong \angle Y \) (as given).

Now, we need to find which leg must be congruent. Based on the options in the image, we need to choose a pair of corresponding legs from the two triangles.

#### Answer:
The correct corresponding leg to be congruent for the HL theorem is \( \overline{BC} \cong \overline{XY} \). This is because these sides correspond to each other as legs in their respective triangles.

### Final Answer: 
The correct option is \( \overline{BC} \cong \overline{XY} \).

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Would you like more details or have any other questions?

Here are 5 related questions to further explore this topic:
1. What other congruence theorems can be used to prove two triangles are congruent?
2. How is the HL Congruence Theorem different from the SAS Congruence Theorem?
3. Can two right triangles be congruent if only their hypotenuses are congruent?
4. What is the significance of the right angle in the HL Congruence Theorem?
5. What are some common mistakes students make when applying the HL Congruence Theorem?

**Tip:** In the HL Theorem, remember that the "leg" referred to must be one of the sides adjacent to the right angle—not the hypotenuse.

Consider right triangles ΔABC and ΔXYZ. Angle B and angle Y are congruent. The hypotenuse of each triangle is AC and XZ, and they are congruent. What other corresponding parts must be congruent to show that ΔABC ≅ ΔXYZ by the HL Congruence Theorem?

Learn how to use the Hypotenuse-Leg (HL) Congruence Theorem to prove that two right triangles ΔABC and ΔXYZ are congruent. We explore how corresponding parts, such as the hypotenuse and legs, must be congruent.

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