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The problem requires proving that triangle \( \triangle ABC \) is congruent to triangle \( \triangle CDA \). Here’s the step-by-step solution:

### Given
1. \( AB \parallel DC \)
2. \( AD \parallel BC \)

### To Prove
\[ \triangle ABC \cong \triangle CDA \]

### Solution

1. **Identify pairs of corresponding angles**:
   - Since \( AB \parallel DC \) and \( AD \parallel BC \), the quadrilateral \( ABCD \) is a parallelogram.
   - In a parallelogram, opposite angles are equal. Therefore:
     - \( \angle BAC = \angle DCA \) (corresponding angles in parallel lines)
     - \( \angle ADB = \angle BCA \) (corresponding angles in parallel lines)

2. **Prove that diagonals bisect each other**:
   - In parallelogram \( ABCD \), diagonals \( AC \) and \( BD \) bisect each other.
   - This gives \( AE = EC \) and \( BE = ED \), where \( E \) is the intersection of \( AC \) and \( BD \).

3. **Use Side-Angle-Side (SAS) Congruence**:
   - In \( \triangle ABC \) and \( \triangle CDA \):
     - **Side**: \( AE = EC \) (since diagonals of a parallelogram bisect each other).
     - **Angle**: \( \angle BAC = \angle DCA \) (as shown in step 1).
     - **Side**: \( BE = ED \) (since diagonals of a parallelogram bisect each other).

   - By SAS congruence, \( \triangle ABC \cong \triangle CDA \).

### Conclusion
Therefore, \( \triangle ABC \cong \triangle CDA \) by the Side-Angle-Side (SAS) criterion.

Would you like more details on any step or have any questions?

### Related Questions
1. How do we prove that a quadrilateral is a parallelogram based on its properties?
2. Why do the diagonals of a parallelogram bisect each other?
3. Can congruent triangles always be found in any parallelogram?
4. How does the Side-Angle-Side (SAS) criterion prove congruence?
5. What other properties can be derived from a parallelogram?

### Tip
When working with parallelograms, remember that opposite angles are equal, and diagonals bisect each other. This often helps in proving triangle congruence within the shape.

Given AB || DC and AD || BC in a parallelogram ABCD, prove that triangle ABC is congruent to triangle CDA.

This solution explains how to prove that triangle ABC is congruent to triangle CDA using the Side-Angle-Side (SAS) criterion. Given that AB is parallel to DC and AD is parallel to BC, the solution utilizes properties of parallelograms to establish triangle congruence.

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