In the given diagram, we see a triangle $\triangle ABC$ with several lines and angles marked, as well as some unknown angles indicated by expressions. Let's analyze and solve for the unknowns.

Observations:

1. There is a triangle formed by $\triangle ABC$ and external lines intersecting it.
2. Several angles are provided:
   • Angle $\angle FIG = 14^\circ$,
   • Angle $\angle GBC = 4x$ (an unknown variable),
   • Angle $\angle GEF = 78^\circ$.

Goal:

Solve for the unknown angle $x$ by using the relationships between angles, such as the sum of angles in a triangle and any supplementary angle rules.

Let's break this down:

1. Using angle properties:
   • The interior angles of a triangle sum up to $180^\circ$.
   • The external angles and parallel lines will provide additional constraints.

Step-by-Step Solution:

• We can look at triangle $EBC$ (where angle $EBC = 4x$, $\angle EGF = 78^\circ$) and find the relationship between these angles to solve for $x$.

I'll proceed with calculations to find the value of $x$.The value of $x$ is $22^\circ$.

Would you like further details on this solution or clarification on how the angles were related?

Here are some related questions for further exploration:

1. How do exterior angles of a triangle relate to the interior angles?
2. Can we apply the same angle sum properties to other polygons?
3. What is the significance of parallel lines in angle determination?
4. How can the concept of transversal lines help solve this problem?
5. Are there any other angle properties (like corresponding or alternate angles) used here?

Tip: When solving geometry problems, always check for triangles and use their angle sum properties. This can simplify the process significantly.