1. **State the problem:** We need to find the size of angle $x$ in triangle $BEG$ given that lines $ABC$ and $DEF$ are parallel and several angles are provided. 2. **Identify known angles and relationships:** - Angle at $D$ is $110^\circ$. - Angle at $E$ is $25^\circ$. - Angle at $B$ is $35^\circ$. - Lines $ABC$ and $DEF$ are parallel, so corresponding and alternate interior angles are equal. 3. **Use parallel lines properties:** Since $ABC \parallel DEF$ and $DE$ is a transversal, the angle at $E$ adjacent to $25^\circ$ is supplementary to $110^\circ$ (since $110^\circ$ is given at $D$ on the same transversal). 4. **Calculate the angle adjacent to $25^\circ$ at $E$:** $$\text{Angle adjacent to } 25^\circ = 180^\circ - 110^\circ = 70^\circ$$ 5. **Calculate angle at $G$ (angle $x$):** Triangle $BEG$ has angles $x$, $35^\circ$, and $25^\circ$ (given). The sum of angles in a triangle is $180^\circ$: $$x + 35^\circ + 25^\circ = 180^\circ$$ $$x + 60^\circ = 180^\circ$$ $$x = 180^\circ - 60^\circ$$ $$x = 120^\circ$$ 6. **Reasoning:** - The sum of angles in a triangle is always $180^\circ$. - Parallel lines create equal corresponding or alternate interior angles. - Supplementary angles add up to $180^\circ$. **Final answer:** $$x = 120^\circ$$