1. **Problem statement:** Given that FH bisects $\angle EFG$, and the segments $EF = 9x + 9$ and $FG = 12x - 3$, find the measure of $\angle EFH$. 2. **Understanding the problem:** Since FH bisects $\angle EFG$, it divides $\angle EFG$ into two equal angles: $\angle EFH = \angle HFG$. 3. **Using the angle bisector theorem:** The angle bisector theorem states that the bisector divides the opposite side into segments proportional to the adjacent sides. Here, the sides $EF$ and $FG$ relate to the segments created by the bisector. 4. **Set the segments equal because FH bisects the angle:** Since FH bisects $\angle EFG$, the two sides are equal in length for the bisected angle, so set $9x + 9 = 12x - 3$. 5. **Solve for $x$:** $$9x + 9 = 12x - 3$$ $$9 + \cancel{9x} = -3 + \cancel{12x}$$ $$9 + 3 = 12x - 9x$$ $$12 = 3x$$ $$x = \frac{12}{3} = 4$$ 6. **Find the lengths of $EF$ and $FG$:** $$EF = 9(4) + 9 = 36 + 9 = 45$$ $$FG = 12(4) - 3 = 48 - 3 = 45$$ 7. **Since $EF = FG$, the bisector divides the angle into two equal parts, so $\angle EFH = \frac{1}{2} \angle EFG$.** 8. **Calculate $\angle EFH$:** If $\angle EFG = 2 \times \angle EFH$, then $$m\angle EFH = \frac{1}{2} m\angle EFG$$ Since the problem does not provide the measure of $\angle EFG$, the best we can say is that $\angle EFH$ is half of $\angle EFG$. **Final answer:** $m\angle EFH = \frac{1}{2} m\angle EFG$ and $x = 4$.