1. **State the problem:** We have an isosceles trapezoid with two equal non-parallel sides. The angles adjacent to these sides are given as $61^\circ$ and $(2x + 15)^\circ$. We need to find the value of $x$. 2. **Important property:** In an isosceles trapezoid, the base angles are equal. Since the trapezoid is isosceles, the angles adjacent to each leg are equal. Therefore, the two given angles must be equal: $$61 = 2x + 15$$ 3. **Solve the equation:** Subtract 15 from both sides: $$61 - 15 = 2x + \cancel{15} - \cancel{15}$$ $$46 = 2x$$ Divide both sides by 2: $$\frac{46}{\cancel{2}} = \frac{2x}{\cancel{2}}$$ $$23 = x$$ 4. **Answer:** The value of $x$ is $23$. This means the angle $(2x + 15)^\circ$ is $2(23) + 15 = 46 + 15 = 61^\circ$, confirming the trapezoid is isosceles with equal base angles.