Identify angle relationships to find x.

Step 1: Identify angle relationships to find $x$. Since line AB is parallel to line CD, the angle $\angle FMA$ and the angle $\angle MNE$ are corresponding angles. Therefore, they are equal. $\angle MNE = \angle FMA = 5x$ Angles $\angle MNE$ and $\angle CNE$ form a linear pair on the straight line CD. Their sum is $180^\circ$. $\angle MNE + \angle CNE = 180^\circ$

Step 2: Substitute the given values and solve for $x$. Substitute $\angle MNE = 5x$ and $\angle CNE = x+78^\circ$ into the equation from Step 1: $5x + (x+78^\circ) = 180^\circ$ $6x + 78^\circ = 180^\circ$ Subtract $78^\circ$ from both sides: $6x = 180^\circ - 78^\circ$ $6x = 102^\circ$ Divide by 6: $x = \frac{102^\circ}{6}$ $x = 17^\circ$

Step 3: Identify angle relationships to find $y$. Since line AB is parallel to line CD, the angle $\angle AMN$ and the angle $\angle DNM$ are alternate interior angles. Therefore, they are equal. $\angle AMN = \angle DNM$ Angles $\angle CNE$ and $\angle DNM$ are vertically opposite angles. Therefore, they are equal. $\angle DNM = \angle CNE = x+78^\circ$ From these relationships, we can write: $\angle AMN = x+78^\circ$

Step 4: Substitute the value of $x$ and solve for $y$. We are given $\angle AMN = 4x+y$. Equate this to the expression from Step 3: $4x+y = x+78^\circ$ Substitute the value of $