Angle Bisector C1C3Bf

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 $x$. 2. **Understand the angle bisector:** Since $PR$ bisects $\angle QPR$, it divides $\angle QPR$ into two equal angles. Therefore, $$m\angle QPR = 2 \times m\angle RPS$$ 3. **Set up the equation:** Given $m\angle QPR = 64^\circ$ and $m\angle RPS = (2x + 10)^\circ$, we write $$64 = 2(2x + 10)$$ 4. **Simplify the equation:** $$64 = 4x + 20$$ 5. **Isolate $x$:** $$64 - 20 = 4x$$ $$44 = 4x$$ 6. **Divide both sides by 4:** $$\cancel{4}x = \frac{44}{\cancel{4}}$$ $$x = 11$$ 7. **Conclusion:** The value of $x$ is $11$. **Final answer:** $x = 11$ (Option A)