1. Analyze Triangle A B C
- We are given that A C=B C , making \(\triangle A B C\) an isosceles triangle.
- In an isosceles triangle, the angles opposite the equal sides are equal. Since the angle opposite side A C is \(\angle A B C=x^{\circ}\) , the angle opposite side B C must also be equal:


\(\angle B A C=x^{\circ}\)


- The sum of the interior angles in any triangle is \(180^{\circ}\) . Therefore, we can find the vertical vertex angle \(\angle A C B\) :


\(\begin{gathered}
\angle A C B=180^{\circ}-(\angle A B C+\angle B A C) \\
\angle A C B=180-2 x
\end{gathered}\)


2. Move Across the Intersection Point C
- Lines B D and A E intersect at point C , meaning \(\angle A C B\) and \( \angle D C E\) are vertical angles and are equal to each other:


\(\angle D C E=\angle A C B=180-2 x\)


2. Move Across the Intersection Point C
- Lines B D and A E $ intersect at point C , meaning \(\angle A C B\) and \(\angle D C E \) are vertical angles and are equal to each other:


\(\angle D C E=\angle A C B=180-2 x\)


3. Analyze Triangle C D E
- We are given that C D=D E , making \(\triangle C D E\) another isosceles triangle.
- The angles opposite these equal sides must be equal. Therefore, the angle opposite side $C D(\angle D E C)$ equals the angle opposite side \(D E(\angle D C E)\) :


\(\angle D E C=\angle D C E=180-2 x\)


- The sum of angles in \(\triangle C D E\) must also equal \(180^{\circ}\) :


\(\angle C D E+\angle D C E+\angle D E C=180^{\circ}\)


- Substitute the expressions we found into the equation:


\(\begin{gathered}
y+(180-2 x)+(180-2 x)=180 \\
y+360-4 x=180
\end{gathered}\)

4. Solve for y


\(\begin{gathered}
y=180-360+4 x \\
y=4 x-180
\end{gathered}\)



Correct Answer: D. \(4 x-180 \)