To determine which two triangles are congruent by the ASA (Angle-Side-Angle) Theorem, let's analyze the given triangles and their markings.

The ASA Theorem states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

1. In triangle $\triangle XYZ$:

   • Angle $\angle XYZ$ and $\angle YXZ$ are marked as congruent to the corresponding angles in another triangle.
   • Side $XY$ is marked with a double line.

2. In triangle $\triangle HJI$:

   • Angle $\angle HJI$ and $\angle JHI$ are marked as congruent.
   • Side $HJ$ is marked with a double line.

3. In triangle $\triangle DCE$:

   • Angle $\angle DCE$ and $\angle CDE$ are marked as congruent.
   • Side $DC$ is marked with a double line.

From these observations, we see that $\triangle XYZ$ and $\triangle HJI$ both have two pairs of congruent angles and the included side marked as congruent. Thus, by the ASA Theorem:

$\triangle XYZ \cong \triangle HJI$

Would you like more details or have any questions?

Related Questions:

1. What are the steps to prove triangle congruence using other theorems like SAS or SSS?
2. How does the ASA Theorem differ from the AAS Theorem in proving triangle congruence?
3. Can the ASA Theorem be used in cases where the side is not included between the angles?
4. Why is it important to identify the correct order of vertices in congruence statements?
5. What are the applications of triangle congruence in real-world scenarios?

Tip: When using the ASA Theorem, always confirm that the side lies between the two angles to meet the requirements of this theorem.