The problem states that $\angle EGH$ is a straight angle, which means $\angle EGH = 180^\circ$. The diagram shows two angles, $\angle FGE = (2x - 18)^\circ$ and $\angle FGH = (4x)^\circ$, that together form the straight angle.

To solve for $m\angle FGH$, follow these steps:

Step 1: Set up the equation.

Since $\angle FGE + \angle FGH = \angle EGH = 180^\circ$, we can write the equation:

$(2x - 18) + 4x = 180$

Step 2: Simplify the equation.

Combine like terms:

$6x - 18 = 180$

Step 3: Solve for $x$.

Add 18 to both sides:

$6x = 198$

Now, divide by 6:

$x = 33$

Step 4: Find $m\angle FGH$.

Substitute $x = 33$ into the expression for $\angle FGH$:

$m\angle FGH = 4x = 4(33) = 132^\circ$

Thus, $m\angle FGH = 132^\circ$.

Would you like more details or have any questions?

Here are 5 related questions:

1. What would $m\angle FGE$ be in this case?
2. How do you verify if $\angle FGE + \angle FGH = 180^\circ$?
3. What is the significance of a straight angle in geometry?
4. How would the problem change if $\angle EGH$ was a different angle, say 150°?
5. How does the linear pair postulate help in solving such problems?

Tip: Remember that angles forming a straight line always add up to $180^\circ$, a key concept in solving angle-related problems.