The theorem that proves $a \parallel b$ in the given diagram is:

B. Same side interior angles of parallel lines cut by a transversal are supplementary.

This is because if the same side interior angles (i.e., angles on the same side of the transversal) add up to 180°, then the lines are proven to be parallel.

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Would you like more details or have any questions about this? Here are 5 related questions to expand your understanding:

1. How do alternate interior angles help in proving lines are parallel?
2. What is the difference between corresponding angles and alternate interior angles?
3. Can parallel lines be proven using exterior angles as well?
4. What does the converse of the same side interior angle theorem state?
5. How can you identify transversal lines in complex diagrams?

Tip: Always remember that supplementary angles add up to 180°, which is key when working with parallel lines and transversals!