In the diagram, DE is parallel to FG. Find the measure of angle DEG in terms of x.

Given that $DE \parallel FG$.
When two parallel chords are intersected by a transversal chord, the alternate interior angles are equal.
Consider EG as a transversal intersecting parallel chords $DE$ and $FG$.
Therefore, $\angle DEG = \angle FGE$ (alternate interior angles).
We are given $\angle G_1 = \angle FGE = x+20^\circ$.
Thus, $\angle DEG = x+20^\circ$.

The size of $\angle DEG$ in terms of $x$ is $\boxed{x+20^\circ}$.