In the figure, $\mathrm{AB}\Vert \mathrm{CD}\Vert \mathrm{EF}, \angle \mathrm{ABG}=110°, \angle \mathrm{GCD}=100°$, and $\angle \mathrm{BGC}=\mathrm{x}°$. What is the value of x?

In the given figure,  $\mathrm{AB}\Vert \mathrm{CD}\Vert \mathrm{EF}$
$\angle \mathrm{ABG}=110°, \angle \mathrm{GCD}=100°$
Also, $\angle \mathrm{ABG}$ and $\angle \mathrm{BGE},\angle \mathrm{GCD}$ and$\angle \mathrm{CGF}$ are pair of interior angles on the same side of the transversal are supplementary.
$\therefore \angle \mathrm{ABG}+\angle \mathrm{BGE}=180°$
$\Rightarrow 110°+\angle \mathrm{BGE}=180°$
$\Rightarrow \angle \mathrm{BGE}=70°$
Similarly,
$\therefore \angle \mathrm{GCD}+\angle \mathrm{CGF}=180°$
$\Rightarrow 100°+\angle \mathrm{CGF}=180°$

$\Rightarrow \angle \mathrm{CGF}=80°$
Now,

$\angle \mathrm{BGE}+\angle \mathrm{BGC}+\angle \mathrm{CGF}=180°$  ...[Angle sum property of a triangle]
$\Rightarrow 70°+\mathrm{x}+80°=180°$

$\Rightarrow \mathrm{x}+150°=180°$
$\Rightarrow \mathrm{x}=30°$