To find the value of $x$ in this geometric configuration, let's analyze the image.

1. The given figure appears to be made up of triangles, including two smaller triangles, $\triangle VWR$ and $\triangle VZR$. We are provided with one angle, $\angle RVW = 57^\circ$, and we need to find $\angle VRY = x^\circ$.

2. Triangle $VWR$ appears to be a straight line, making $\triangle VWR$ a straight angle where the sum of the interior angles is $180^\circ$. Since $\angle RVW = 57^\circ$, we can find $\angle WRV$ as: $\angle WRV = 180^\circ - 57^\circ = 123^\circ$

3. We now focus on $\triangle VRY$. Since we know that $\angle WRV = 123^\circ$, $x$ is an exterior angle for $\triangle VRY$, and the exterior angle is equal to the sum of the two non-adjacent interior angles. Therefore: $x = 57^\circ$

Thus, $x = 57^\circ$.

Let me know if you would like further explanation or have additional questions!

Here are five follow-up questions:

1. What are the properties of exterior angles in a triangle?
2. How do you calculate the sum of interior angles in any polygon?
3. What would happen to $x$ if angle $\angle RVW$ was different?
4. How can you verify the solution using other angles in the figure?
5. Can you apply this method to other types of triangles?

Tip: When solving for unknown angles, remember that the exterior angle of a triangle is always equal to the sum of the opposite interior angles.