1. **State the problem:** We have a kite-shaped quadrilateral HIJK with angles at I and K given as $123^\circ$ and $105^\circ$ respectively. We need to find the measure of angle $\angle H$. 2. **Recall kite properties:** In a kite, two pairs of adjacent sides are equal. Also, the angles between unequal sides are equal. Here, $HI = HK$ and $IJ = JK$. 3. **Sum of interior angles:** The sum of interior angles in any quadrilateral is $360^\circ$. So, $$m\angle H + m\angle I + m\angle J + m\angle K = 360^\circ$$ 4. **Use kite angle property:** The angles between the pairs of equal sides are equal. Since $HI = HK$, angles at $J$ and $I$ are equal, so $$m\angle J = m\angle I = 123^\circ$$ 5. **Substitute known values:** $$m\angle H + 123^\circ + 123^\circ + 105^\circ = 360^\circ$$ 6. **Simplify:** $$m\angle H + 351^\circ = 360^\circ$$ 7. **Solve for $m\angle H$:** $$m\angle H = 360^\circ - 351^\circ = 9^\circ$$ **Final answer:** $$\boxed{9^\circ}$$