1. **State the problem:** We are given a triangle with angles $115^\circ$, $(4k + 5)^\circ$, and $(6k + 10)^\circ$. We need to find the value of $k$. 2. **Recall the Triangle Angle Sum Theorem:** The sum of the interior angles of a triangle is always $180^\circ$. So, $$115 + (4k + 5) + (6k + 10) = 180$$ 3. **Set up the equation and simplify:** $$115 + 4k + 5 + 6k + 10 = 180$$ Combine like terms: $$115 + 5 + 10 + 4k + 6k = 180$$ $$130 + 10k = 180$$ 4. **Solve for $k$:** Subtract 130 from both sides: $$10k = 180 - 130$$ $$10k = 50$$ Divide both sides by 10: $$k = \frac{50}{10} = 5$$ 5. **Answer:** The value of $k$ is $5$. This means the angles are $115^\circ$, $(4(5) + 5)^\circ = 25^\circ$, and $(6(5) + 10)^\circ = 40^\circ$, which sum to $180^\circ$ confirming the solution.