1. **State the problem:** We have a kite DEFG with angles $m\angle DEF = (12x - 16)^\circ$, $m\angle EFH = (3x - 1)^\circ$, and $m\angle DGF = 74^\circ$. We need to find $m\angle GFE$. 2. **Recall kite properties:** In a kite, two pairs of adjacent sides are equal, and one diagonal bisects the other at right angles. Also, the angles between unequal sides are equal. 3. **Identify angles:** $\angle EFH$ is part of the diagonal intersection inside the kite, so $\angle EFH = \angle GFH = (3x - 1)^\circ$ because the diagonal bisects the angle. 4. **Use triangle DEF:** The sum of angles in triangle DEF is $180^\circ$: $$ (12x - 16) + (3x - 1) + m\angle DFE = 180 $$ 5. **Simplify:** $$ 12x - 16 + 3x - 1 + m\angle DFE = 180 $$ $$ 15x - 17 + m\angle DFE = 180 $$ 6. **Solve for $m\angle DFE$:** $$ m\angle DFE = 180 - 15x + 17 = 197 - 15x $$ 7. **Use triangle DGF:** Given $m\angle DGF = 74^\circ$, and $m\angle DFG = m\angle DFE = 197 - 15x$ (since $\angle DFE$ and $\angle DFG$ are equal in kite), sum of angles in triangle DGF is: $$ 74 + (197 - 15x) + m\angle GFE = 180 $$ 8. **Simplify:** $$ 271 - 15x + m\angle GFE = 180 $$ 9. **Solve for $m\angle GFE$:** $$ m\angle GFE = 180 - 271 + 15x = 15x - 91 $$ 10. **Find $x$ using kite angle property:** The sum of angles at point F is $360^\circ$, so: $$ (12x - 16) + (3x - 1) + (15x - 91) + 74 = 360 $$ 11. **Simplify:** $$ 12x - 16 + 3x - 1 + 15x - 91 + 74 = 360 $$ $$ 30x - 34 = 360 $$ 12. **Solve for $x$:** $$ 30x = 394 $$ $$ x = \frac{394}{30} = \frac{197}{15} \approx 13.13 $$ 13. **Calculate $m\angle GFE$:** $$ m\angle GFE = 15x - 91 = 15 \times \frac{197}{15} - 91 = 197 - 91 = 106^\circ $$ **Final answer:** $m\angle GFE = 106^\circ$.