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The problem asks to prove that triangles \( \triangle ABC \cong \triangle CDA \) under the given conditions:

- \( AD \cong BC \) (segments \( AD \) and \( BC \) are congruent).
- \( AD \parallel BC \) (segments \( AD \) and \( BC \) are parallel).

### Proof Outline:

To prove that \( \triangle ABC \cong \triangle CDA \), we can use the **ASA (Angle-Side-Angle) Congruence Theorem**. Let's break down the steps:

1. **Step 1: Given**  
   As stated in the problem, we are given that \( AD \cong BC \) and \( AD \parallel BC \).

2. **Step 2: Corresponding Angles**  
   Since \( AD \parallel BC \) and \( AB \) is a transversal, we know that:
   - \( \angle DAB \cong \angle BCA \) by the Corresponding Angles Postulate (when a transversal crosses parallel lines, corresponding angles are congruent).

3. **Step 3: Vertical Angles**  
   The triangles share a common angle at \( A \), so:
   - \( \angle BAC \cong \angle DAC \) by the Vertical Angles Theorem.

4. **Step 4: Prove Congruence (ASA)**  
   We now have two pairs of congruent angles and one pair of congruent sides:
   - \( \angle DAB \cong \angle BCA \) (from Step 2),
   - \( AD \cong BC \) (Given),
   - \( \angle BAC \cong \angle DAC \) (from Step 3).

Thus, by the **ASA Congruence Theorem**, \( \triangle ABC \cong \triangle CDA \).

### Summary of Proof:

| Step  | Statement                          | Reason                                          |
|-------|------------------------------------|-------------------------------------------------|
| 1     | \( AD \cong BC \) and \( AD \parallel BC \) | Given                                           |
| 2     | \( \angle DAB \cong \angle BCA \)  | Corresponding Angles Postulate                  |
| 3     | \( \angle BAC \cong \angle DAC \)  | Vertical Angles Theorem                         |
| 4     | \( \triangle ABC \cong \triangle CDA \) | ASA (Angle-Side-Angle) Congruence Theorem       |

Would you like to go into more detail or have any further questions? 

### Related Questions:
1. Can we use the SAS (Side-Angle-Side) theorem in this proof instead of ASA?
2. How do corresponding angles work in a transversal situation?
3. What are some other ways to prove triangle congruence?
4. Could we have used another postulate like AAS (Angle-Angle-Side) for this proof?
5. Why are quadrilateral properties not allowed in this proof?

### Tip:
Always check for parallel lines and transversals in geometric proofs, as they often help establish congruent angles.

Given AD ≅ BC and AD ∥ BC, prove ΔABC ≅ ΔCDA.

Learn how to prove that ΔABC ≅ ΔCDA using the ASA (Angle-Side-Angle) congruence theorem. This geometric proof involves congruent sides, parallel lines, and corresponding angles. Suitable for high school students in Grades 9-12.

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