To understand the proof of the theorem \( a \parallel b \) based on the given image, we have the scenario set up with angle measurements and parallel lines.

### Steps to Prove \( a \parallel b \):

1. **Identify Angles**: Look for corresponding or alternate interior angles between the two lines and the transversal.

2. **Angle Relationships**:
- From the diagram, if \(\angle A_1 = \angle B_2\) or \(\angle A_2 + \angle B_1 = 180^\circ\), you can conclude that lines \( a \) and \( b \) are parallel.

3. **Use Transversal Properties**:
- If two lines are cut by a transversal and the corresponding angles are equal or if the sum of the interior angles on the same side is supplementary, the lines are parallel.

4. **Conclusion**: Based on angle measurements, conclude that lines \( a \) and \( b \) are parallel (\( a \parallel b \)).

Make sure to refer to the specific angles and their relationships to solidify the proof mathematically.