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Determine if the two triangles are congruent.

If the triangles are congruent, write a congruence statement (such as 
) and give the congruence theorem (such as 
).

If the triangles are not congruent, or if there is not enough information to determine congruence, write “cannot be determined” and explain why not.

Looking at the diagram of triangles \( \triangle ABD \) and \( \triangle ACD \), we can analyze the given information to determine if the two triangles are congruent.

### Key Observations:
1. **Side \( AB = AC \)**: Both sides are marked with the same symbol, indicating they are congruent.
2. **Side \( BD = CD \)**: These sides are also marked congruent.
3. **Angle \( \angle BDA = \angle CDA \)**: These angles are marked as congruent (corresponding angle marks).

### Conclusion:
We have the following congruent elements:
- **Two sides (AB = AC and BD = CD)**
- **The included angle ( \( \angle BDA = \angle CDA \) )**

This information fits the **SAS (Side-Angle-Side)** congruence theorem, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

### Congruence Statement:
\[
\triangle ABD \cong \triangle ACD
\]

### Theorem:
The triangles are congruent by **SAS** (Side-Angle-Side).

Let me know if you need further details or have any questions!

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Here are some related questions:
1. How do we verify if angles in two triangles are congruent?
2. What is the importance of the "included angle" in the SAS congruence theorem?
3. Can the two triangles be congruent using another theorem like SSS or ASA?
4. What would happen if the included angle were not given? Would we still be able to prove congruence?
5. Are there other properties of congruent triangles that could help verify the solution further?

**Tip:** When identifying congruent triangles, always check for corresponding parts and whether enough information satisfies one of the congruence theorems (SAS, SSS, ASA, AAS).

Determine if the two triangles are congruent. If the triangles are congruent, write a congruence statement and give the congruence theorem. If the triangles are not congruent, or if there is not enough information to determine congruence, write 'cannot be determined' and explain why not.

Learn how to determine the congruence of two triangles using the SAS (Side-Angle-Side) theorem. This problem involves analyzing the given sides and angles of triangles ABD and ACD to prove their congruence. Suitable for students in Grades 8-10.

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