1. **State the problem:** We need to find the value of $t$ in a triangle where the angles are given as $32^\circ$, $t + 45^\circ$, and the bottom vertex angle is split into two parts: $10c - 16^\circ$ and $3t - 41^\circ$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of a triangle is always $180^\circ$. 3. **Express the total angle at the bottom vertex:** Since the bottom vertex angle is split into two parts, its total angle is: $$ (10c - 16) + (3t - 41) = 10c - 16 + 3t - 41 = 10c + 3t - 57 $$ 4. **Write the equation for the sum of angles:** $$ 32 + (t + 45) + (10c + 3t - 57) = 180 $$ 5. **Simplify the equation:** $$ 32 + t + 45 + 10c + 3t - 57 = 180 $$ $$ (32 + 45 - 57) + t + 3t + 10c = 180 $$ $$ 20 + 4t + 10c = 180 $$ 6. **Isolate terms:** $$ 4t + 10c = 180 - 20 $$ $$ 4t + 10c = 160 $$ 7. **Solve for $t$ in terms of $c$:** $$ 4t = 160 - 10c $$ $$ t = \frac{160 - 10c}{4} $$ 8. **Simplify the fraction:** $$ t = \frac{\cancel{160} - \cancel{10}c}{\cancel{4}} = 40 - \frac{5c}{2} $$ **Final answer:** $$ t = 40 - \frac{5c}{2} $$ Since $c$ is not given, $t$ is expressed in terms of $c$.