1. **Problem statement:** Quadrilateral DEFG is a kite with angles at vertices E and G given as $55^\circ$ and $89^\circ$ respectively. We need to find the measure of angle $F$. 2. **Properties of a kite:** A kite has two pairs of adjacent sides equal and its opposite angles between unequal sides are equal. Also, the sum of interior angles in any quadrilateral is $360^\circ$. 3. **Sum of angles formula:** $$m\angle D + m\angle E + m\angle F + m\angle G = 360^\circ$$ 4. **Given values:** $$m\angle E = 55^\circ, \quad m\angle G = 89^\circ$$ 5. **Using kite property:** Angles $D$ and $F$ are equal because they are opposite angles between the pairs of equal sides. 6. **Let:** $$m\angle D = m\angle F = x$$ 7. **Substitute into sum of angles:** $$x + 55 + x + 89 = 360$$ 8. **Simplify:** $$2x + 144 = 360$$ 9. **Isolate $x$:** $$2x = 360 - 144$$ $$2x = 216$$ 10. **Divide both sides by 2:** $$\cancel{2}x = \cancel{2}108$$ $$x = 108$$ 11. **Conclusion:** $$m\angle F = 108^\circ$$