To prove that \( a \parallel b \) in the given geometric figure, we can use the concept of corresponding angles or alternate interior angles, depending on the configuration.

1. **Identify Angles:**
- In the first figure, you have two lines intersected by a transversal. Denote the angles formed at the intersection.
- For the given information, if \( \angle A \) equals \( 50^\circ \) and \( \angle B \) equals \( 130^\circ \).

2. **Using Angle Relationships:**
- If \( \angle A \) and \( \angle B \) are corresponding angles or alternate interior angles, then:
\[
\angle A + \angle B = 180^\circ
\]
- Confirm that \( 50^\circ + 130^\circ = 180^\circ \), which shows these angles are supplementary.

3. **Conclude Parallelism:**
- Since the angles are supplementary and the lines are intersected by a transversal, by the Converse of the Alternate Interior Angles Theorem, we conclude that:
\[
a \parallel b
\]

This reasoning shows that if the angles satisfy the conditions for parallel lines, then \( a \) and \( b \) are indeed parallel.