Find the values of x and y

We know that $(5x + 2y - 25) + z = 180$. 

Solving for z, we find that $z = 205 - 5x - 2y$

Using, alternate interior angles, we know that $2y + x + y + 205 - 5x - 2y = 180$

Simplifying, we find that $25 = 4x - y$

Because we want $L_4$ to be parallel to $L_3$, the angles must be equal, meaning:  $2y = 4x + 10$

We now have a system of equations to solve: $2y = 4x +10$ and $25 = 4x - y$

Solving, we find $\color{brown}\boxed{x =15, \space y = 35}$

If L3  is parallel to L4, then, by corresponding angles

4x + 10  = 2y       rearrange   as

y  = 2x + 5         (1)

And if L1 is parallel to L2, then, by the property of supplementary angles

5x + 2y - 25 + 2y + x + y    =  180       rearrange as

6x + 5y  =  205     (2)

Sub (1) into (2)

6x + 5 ( 2x + 5)   =  205

6x + 10x + 25   = 205

16x  =  180

x =  180 / 16   =   45/4   = 11.25

And 

y = 2 (11.25) + 5 =   27.5