https://file.aiquickdraw.com/mathbot/file/1726589412334bfv7whhr.jpg

The image you've provided shows two right-angled triangles: △ABC on the left and △XYZ on the right. We are tasked with determining if these triangles are congruent and, if so, identifying the appropriate congruence rule (AAS, SAS, HL, SSS, or whether they are not congruent).

### Analyzing the triangles:
- **Triangle ABC**:
  - Right angle at \( \angle ACB \).
  - Two sides are marked as equal: \( AC \cong XY \) and \( AB \cong XZ \).
  
- **Triangle XYZ**:
  - Right angle at \( \angle XYZ \).
  - Two sides marked as equal: \( XY \cong AC \) and \( XZ \cong AB \).

### Applying the congruence criteria:
- **Right angles**: Both triangles have right angles at \( \angle ACB \) and \( \angle XYZ \).
- **Hypotenuses**: The hypotenuses \( AB \) and \( XZ \) are marked as congruent.
- **Legs**: The legs \( AC \) and \( XY \) are also marked as congruent.

This satisfies the **Hypotenuse-Leg (HL)** congruence theorem, which applies to right triangles. Therefore, **the two triangles are congruent by HL**.

### Conclusion:
The triangles are congruent by the **Hypotenuse-Leg (HL) congruence theorem**.

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Would you like further details or have any questions?

Here are 5 related questions for further exploration:
1. What is the Hypotenuse-Leg (HL) theorem, and why does it apply only to right triangles?
2. Can other congruence criteria like AAS, SAS, or SSS apply to right triangles?
3. How does the HL theorem differ from the SSS theorem?
4. If one leg of the triangle was not congruent, could the triangles still be congruent under a different criterion?
5. How can we prove triangle congruence using geometric transformations (e.g., reflection or rotation)?

**Tip**: Always check for right angles in triangles as they often lead to simpler congruence tests like the HL theorem.

Analyze two right-angled triangles and determine their congruence using the Hypotenuse-Leg (HL) theorem. This detailed solution explains why the triangles are congruent, exploring core concepts in triangle geometry for students in Grades 8-10.

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