Question: User: 21. In the figure shown, PR bisects \u2220QPR, m\u2220QPR = 64\u00b0, and m\u2220RPS = (2x + 10)\u00b0. What is the value of x?\n\nA. 11\nB. 21\nC. 27\nD. 53\n\nGraph/diagram: A point P at the lower left has three rays extending upward/right to Q, to the upper-right toward R, and horizontally right to S; PR is the middle ray and is shown bisecting \u2220QPS. position_hint = top-left
1. **State the problem:** We are given that $PR$ bisects $\angle QPR$, the measure of $\angle QPR$ is $64^\circ$, and the measure of $\angle RPS$ is $(2x + 10)^\circ$. We need to find the value of $x$.\n\n2. **Understand the angle bisector:** Since $PR$ bisects $\angle QPR$, it divides $\angle QPR$ into two equal angles. Therefore, each of the two angles formed by $PR$ is half of $64^\circ$, so $m\angle QPR = m\angle QPR_1 + m\angle QPR_2 = 64^\circ$ and $m\angle QPR_1 = m\angle QPR_2 = 32^\circ$.\n\n3. **Identify the angles:** The problem states $m\angle RPS = (2x + 10)^\circ$. Since $PR$ bisects $\angle QPR$, $\angle RPS$ is one of the two equal angles formed by the bisector. So, $m\angle RPS = 32^\circ$.\n\n4. **Set up the equation:** $2x + 10 = 32$\n\n5. **Solve for $x$:**\n$2x = 32 - 10$\n$2x = 22$\n$x = \frac{22}{2}$\n$x = 11$\n\n6. **Conclusion:** The value of $x$ is $11$.\n\n**Final answer:** $x = 11$ (Option A)