1. **Problem Statement:** Quadrilateral PQRS is inscribed in circle A. Given that $m\angle P = 92^\circ$ and $m\angle Q = 92^\circ$, determine which statement about angles $R$ and $S$ is necessarily true. 2. **Key Property:** In a cyclic quadrilateral (one inscribed in a circle), opposite angles are supplementary. This means: $$m\angle P + m\angle R = 180^\circ$$ $$m\angle Q + m\angle S = 180^\circ$$ 3. **Calculate $m\angle R$:** Using the property for opposite angles, $$m\angle R = 180^\circ - m\angle P = 180^\circ - 92^\circ = 88^\circ$$ 4. **Calculate $m\angle S$:** Similarly, $$m\angle S = 180^\circ - m\angle Q = 180^\circ - 92^\circ = 88^\circ$$ 5. **Analyze the statements:** - A. $m\angle R = m\angle S$ is true since both are $88^\circ$. - B. $m\angle R + m\angle S = 88^\circ + 88^\circ = 176^\circ \neq 180^\circ$ (false). - C. $m\angle R = \frac{1}{2} m\angle S$ means $88^\circ = 44^\circ$ (false). - D. $m\angle R + m\angle S = m\angle P + m\angle Q$ means $176^\circ = 184^\circ$ (false). **Final answer:** Statement A is necessarily true.