1. **Stating the problem:** We have a trapezoid with bases of lengths $a$ (top) and 46 (bottom). The lower left angle is $60^\circ$, the lower right angle is $55^\circ$, and a segment inside the trapezoid parallel to the bases has length 31. We need to find the value of angle $w^\circ$ adjacent to the $60^\circ$ angle. 2. **Understanding the trapezoid and angles:** In a trapezoid, the consecutive angles between a leg and the bases are supplementary if the bases are parallel. The segment inside parallel to the bases creates similar triangles. 3. **Using angle sum properties:** The lower left angle is $60^\circ$, and the adjacent angle $w^\circ$ plus $60^\circ$ should sum to $180^\circ$ because they are on a straight line: $$w + 60 = 180$$ 4. **Solving for $w$:** $$w = 180 - 60$$ $$w = 120$$ 5. **Conclusion:** The value of $w$ is $120^\circ$. **Final answer:** $$\boxed{120}$$