1. **State the problem:** We have a triangle with interior angles $35^\circ$, $(5x + 7)^\circ$, and $(9x - 2)^\circ$. We need to find $x$ and the measure of the angle $(5x + 7)^\circ$. 2. **Formula:** The sum of interior angles in any triangle is always $180^\circ$. So, $$35 + (5x + 7) + (9x - 2) = 180$$ 3. **Simplify the equation:** $$35 + 5x + 7 + 9x - 2 = 180$$ $$35 + 7 - 2 + 5x + 9x = 180$$ $$40 + 14x = 180$$ 4. **Isolate $x$:** $$14x = 180 - 40$$ $$14x = 140$$ 5. **Divide both sides by 14:** $$x = \cancel{\frac{14x}{14}} = \frac{140}{\cancel{14}}$$ $$x = 10$$ 6. **Find the measure of the indicated angle $m\angle X = (5x + 7)^\circ$:** $$m\angle X = 5(10) + 7 = 50 + 7 = 57^\circ$$ **Final answers:** $$x = 10$$ $$m\angle X = 57^\circ$$