To determine the size of angle $x$ in triangle $BEG$, we can proceed by using angle properties in parallel lines and triangles. Here's the solution step-by-step:

1. Identify the Corresponding Angles: Since $ABC$ and $DEF$ are parallel lines:

   • The angle $\angle B$ (34.8°) and $\angle GEF$ (25.6°) will allow us to work out angle $\angle EBG$.

2. Determine $\angle EBG$:

   • In triangle $BEG$, the sum of interior angles is 180°.
   • We know two angles:
     • $\angle EBG = 34.8°$
     • $\angle BEG = 25.6°$
   • Use the sum of angles in a triangle: [ \angle EBG + \angle BEG + x =