1. **State the problem:** We are given the exterior angles of a convex hexagon as expressions involving $t$: $5t - 38^\circ$, $7t - 43^\circ$, $36^\circ$, $t + 20^\circ$, $6t - 24^\circ$, and $13t - 39^\circ$. We need to find the value of $t$. 2. **Recall the key property:** The sum of the exterior angles of any convex polygon is always $360^\circ$. 3. **Set up the equation:** $$ (5t - 38) + (7t - 43) + 36 + (t + 20) + (6t - 24) + (13t - 39) = 360 $$ 4. **Combine like terms:** $$ 5t + 7t + t + 6t + 13t - 38 - 43 + 36 + 20 - 24 - 39 = 360 $$ $$ (5t + 7t + t + 6t + 13t) + (-38 - 43 + 36 + 20 - 24 - 39) = 360 $$ $$ 32t + (-88) = 360 $$ 5. **Isolate $t$:** $$ 32t = 360 + 88 $$ $$ 32t = 448 $$ 6. **Solve for $t$:** $$ t = \frac{448}{32} $$ $$ t = \cancel{\frac{448}{32}} = 14 $$ **Final answer:** $t = 14^\circ$