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The problem asks which statement belongs in the missing blank to complete the proof of triangle congruence between \( \triangle BCD \) and \( \triangle DAB \). Let's analyze it step by step based on the given diagram and statements:

### Key information:
- Line \( AB \parallel DC \), and \( DB \) is a transversal. Therefore, angles \( \angle CDB \) and \( \angle ADB \) are alternate interior angles and congruent.
- Side \( DB = BD \) since it's the same segment.
- Angle \( A = C \) as they are both right angles.
- By the Angle-Side-Angle (ASA) congruence theorem, \( \triangle BCD \cong \triangle DAB \).

The missing step likely involves confirming that the triangles have corresponding sides congruent or parallel to set up the ASA congruence.

### Answer Choices:
1. **"Side DA is parallel to side BC by the converse of the Same Side Interior Angle Theorem"**: This is correct. Since \( AB \parallel DC \) and \( DB \) is a transversal, the same logic of alternate interior angles applies, showing that \( DA \parallel BC \). This completes the proof needed for triangle congruence.

The correct statement to fill in the blank is the first option.

### Would you like further details, or do you have questions about this?

Here are five related questions to expand on this concept:
1. What is the Angle-Side-Angle (ASA) theorem?
2. How does the Alternate Interior Angle Theorem apply to parallel lines and transversals?
3. Why is \( \angle A \) congruent to \( \angle C \) in this diagram?
4. Can we use the Side-Angle-Side (SAS) theorem instead of ASA here?
5. How do you differentiate between the converse and the direct form of a theorem in geometry?

**Tip**: Always look for key parallel line relationships when working with angle congruences in triangle proofs.

Han wrote a proof that triangle BCD is congruent to triangle DAB. Han's proof is incomplete.

This problem explores the proof of triangle BCD congruent to triangle DAB using the Angle-Side-Angle theorem. Learn how to use parallel lines, alternate interior angles, and the Same Side Interior Angle Theorem to complete the proof.

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