1. **Problem statement:** We have an equilateral triangle QPR, and we are given that angle PRS equals $2x - 10^\circ$. We need to find the value of $x$. 2. **Important properties:** In an equilateral triangle, all sides are equal, and all interior angles are equal to $60^\circ$. 3. **Step-by-step solution:** - Since QPR is equilateral, angle QPR = angle PRQ = angle PQR = $60^\circ$. - Point S lies on the extension or inside the triangle such that angle PRS is given. - Because angle PRS is related to the triangle's angles, and since PRS includes angle PRQ or is supplementary to it, we set: $$2x - 10 = 60$$ - Solve for $x$: $$2x = 60 + 10$$ $$2x = 70$$ $$x = \frac{70}{2} = 35$$ 4. **Answer:** The value of $x$ is $35$. This uses the property that all angles in an equilateral triangle are $60^\circ$, and the given angle PRS equals $2x - 10^\circ$ which must be $60^\circ$.