To prove that lines \( AB \) and \( CD \) are parallel, we can use the concept of corresponding angles. Here's how to approach the proof:

1. **Identify the Angles:**
- Given \( \angle ABM = 30^\circ \)
- Given \( \angle MDC = 60^\circ \)

2. **Use the Transversal:**
- The line \( BM \) serves as a transversal to the lines \( AB \) and \( CD \).

3. **Calculate the Angles:**
- Since \( \angle ABM = 30^\circ \), the angle \( \angle MBM \) is complementary to \( \angle ABM \):
\[
\angle MBC = 180^\circ - 30^\circ = 150^\circ
\]
- For angle \( \angle MDC \), we already have it as \( 60^\circ \).

4. **Determine Relationship:**
- Since \( \angle MBC = 150^\circ \) and \( \angle MDC = 60^\circ \):
\[
\angle MBC + \angle MDC = 180^\circ
\]
This confirms that these angles are supplementary.

5. **Conclusion:**
- Since the corresponding angles along the transversal \( BM \) formed with \( AB \) and \( CD \) add up to \( 180^\circ \), we can conclude:
\[
AB \parallel CD
\]

Thus, we have demonstrated that line \( AB \) is parallel to line \( CD \) using angle relationships related to a transversal.