We are given that $\overrightarrow{FH}$ bisects $\angle EFG$, and that the measure of $\angle HFG = 18^\circ$. Since $\overrightarrow{FH}$ bisects $\angle EFG$, it divides the angle into two equal parts.

Thus, $m \angle EFG = 2 \times m \angle HFG$.

Given that $m \angle HFG = 18^\circ$, we can calculate $m \angle EFG$ as follows:

$m \angle EFG = 2 \times 18^\circ = 36^\circ$

Therefore, the measure of $\angle EFG$ is $36^\circ$.

Would you like further details or have any additional questions?

Here are 5 related questions for practice:

1. If $\overrightarrow{FH}$ bisects $\angle ABC$ and $m \angle HBC = 25^\circ$, what is $m \angle ABC$?
2. If a bisector divides an angle into two parts, and one part is $45^\circ$, what is the full angle?
3. What is the measure of an angle if it is bisected and each resulting angle is $30^\circ$?
4. If an angle is bisected and the measure of one part is $22^\circ$, what is the measure of the full angle?
5. Can an angle bisector divide an obtuse angle? Why or why not?

Tip: Always remember, an angle bisector divides an angle into two equal parts!