To prove that \( a \parallel b \) based on the given information, we can utilize the properties of angles and parallel lines.

Given:
- \( A_4 + A_4 + B_4 = 288^\circ \)
- \( A_1 = \frac{2}{3} A_2 \)

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

1. **Angle Relationships**: Since angles \( A_4 \) and \( B_4 \) are consecutive interior angles along the transversal \( c \), if the sum of these angles alongside the angle \( A_4 \) is \( 288^\circ \), we can analyze the supplementary angles formed by the parallel lines cut by a transversal.

2. **Calculate \( A_4 + B_4 \)**:
- From the equation, we know \( A_1 + A_2 + B_4 = 288^\circ \).
- Rearranging gives \( A_4 + B_4 = 288^\circ - A_4 \).

3. **Declare Alternate Interior Angles**: If \( A_1 \) can be recalculated based on \( A_2 \) (which is given as \( A_1 = \frac{2}{3} A_2 \)), substitute this into the equation.

4. **Conclude Angle Conditions**: If \( A_4 + B_4 = 288^\circ \) where supplementary relationships hold, it implies that the corresponding angles formed at the transversal are equal or supplementary to one another.

5. **Apply the converse of the Alternate Interior Angles Theorem**: If the converse holds (the corresponding angles are equal), we conclude:
- Since \( A_4 + B_4 = 288^\circ \), this implies that the transversal \( c \) must create equal angles with the lines \( a \) and \( b \).

### Conclusion:
Thus, since \( A_4 + B_4 = 288^\circ \) demonstrates that the angles maintain the necessary conditions, it follows:

\[
a \parallel b
\]

This concludes the proof required.