In the diagram, $A, B, D, F$, and $G$ lie on a vertical line, $\triangle B C D$ is right-angled at $C$, and $\triangle D E F$ is rightangled at $E$. Also, $\angle A B C=x^{\circ}, \angle C D E=80^{\circ}$, and $\angle E F G=y^{\circ}$. What is the value of $x+y$ ?
(A) 250
(B) 260
(C) 270
(D) 280
(E) 290

![](https://cdn.mathpix.com/cropped/2024_04_20_6ed09463f225f8ba1f07g-004.jpg?height=585&width=223&top_left_y=217&top_left_x=1382)
Since $\angle A B D=180^{\circ}$ and $\angle A B C=x^{\circ}$, then $\angle C B D=180^{\circ}-x^{\circ}$.

Since the measures of the angles in $\triangle B C D$ add to $180^{\circ}$, then

$$
\angle B D C=180^{\circ}-\left(180^{\circ}-x^{\circ}\right)-90^{\circ}=x^{\circ}-90^{\circ}
$$

Similarly, $\angle G F D=180^{\circ}$ and $\angle F D E=y^{\circ}-90^{\circ}$. Finally, $\angle B D F=180^{\circ}$ and so

$$
\begin{aligned}
\angle B D C+\angle C D E+\angle F D E & =180^{\circ} \\
\left(x^{\circ}-90^{\circ}\right)+80^{\circ}+\left(y^{\circ}-90^{\circ}\right) & =180^{\circ} \\
x+y-100 & =180
\end{aligned}
$$

and so $x+y=280$.

ANSWER: (D)