This problem involves finding the value of $x$ in the context of vertical angles. Vertical angles are equal, so we can set the two angle expressions equal to each other.

Step-by-Step Solution

Given: $(3x - 14)^\circ = [4(x - 9)]^\circ$

1. Expand the right side: $3x - 14 = 4x - 36$

2. Rearrange terms to isolate $x$: Subtract $3x$ from both sides: $-14 = x - 36$

3. Solve for $x$: Add 36 to both sides: $x = 22$

Answer

$x = 22$

Would you like further details on the steps?

Here are some related questions:

1. What are vertical angles, and why are they equal?
2. How do you solve linear equations with variables on both sides?
3. What is the definition of supplementary angles?
4. How do you simplify expressions with parentheses?
5. What are alternate interior angles, and how do they relate to this problem?

Tip: When solving equations, always check your solution by substituting back into the original expressions to verify they are equal.