Let's break down the problem based on the provided diagram and the given information:

• Lines $l_1$ and $l_2$ are parallel.
• $\angle BAG = (2x - 112)^\circ$
• $\angle ACK = 118^\circ$
• $\angle ABC = (x - 25)^\circ$

Step-by-step Approach:

1. Identify corresponding angles: Since $l_1$ and $l_2$ are parallel and we have a transversal (line $AB$), we can use the properties of corresponding angles and alternate interior angles.

2. Use linear pairs:

   • $\angle BAG$ and $\angle ABC$ form a linear pair at point $A$, which means: $\angle BAG + \angle ABC = 180^\circ$
   • Substituting the given values: $(2x - 112) + (x - 25) = 180^\circ$ Simplifying this equation will help us find $x$.

Let me solve this now.The value of $x$ is $\frac{317}{3}$, which is approximately $105.67^\circ$.

Thus, $x = 105.67^\circ$.

Would you like further clarification or details on the solution? Here are five related questions for further exploration:

1. How do we use alternate interior angles when parallel lines are involved?
2. What is the relationship between corresponding angles and parallel lines?
3. How would the solution change if one of the angles was altered?
4. What would be the angle sum for a triangle in this diagram?
5. How can we verify the consistency of the solution using other angle properties?

Tip: Remember that when dealing with parallel lines and a transversal, corresponding angles are always equal.