1. **State the problem:** We are given a kite-shaped quadrilateral with two angles labeled as $(5x - 1)^\circ$ and $8x^\circ$. We need to find the value of $x$. 2. **Recall kite properties:** In a kite, two pairs of adjacent sides are equal, and the angles between unequal sides are equal. Also, the sum of interior angles in any quadrilateral is $360^\circ$. 3. **Set up the equation:** The two given angles are adjacent angles in the kite. Since the kite is symmetric along its diagonals, these two angles are supplementary (sum to $180^\circ$). 4. **Write the equation:** $$ (5x - 1) + 8x = 180 $$ 5. **Simplify the equation:** $$ 5x - 1 + 8x = 180 $$ $$ 13x - 1 = 180 $$ 6. **Add 1 to both sides:** $$ 13x - \cancel{1} + \cancel{1} = 180 + 1 $$ $$ 13x = 181 $$ 7. **Divide both sides by 13:** $$ \frac{13x}{\cancel{13}} = \frac{181}{13} $$ $$ x = \frac{181}{13} \approx 13.923 $$ 8. **Final answer:** $$ x \approx 13.92 $$