2023 Pascal Problem 18

In the diagram, A, B, D, F, and G lie on a vertical line, \triangle BCD is right-angled at C, and \triangle DEF is right-angled at E. Also, \angle ABC = x^\circ, \angle CDE = 80^\circ, and \angle EFG = y^\circ. What is the value of x + y?

\textbf{(A)}\ 250\quad \textbf{(B)}\ 260\quad \textbf{(C)}\ 270\quad \textbf{(D)}\ 280\quad \textbf{(E)}\ 290

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Pascal contests here.