1. **State the problem:** Given that $m\angle SQT = 30^\circ$ and $m\angle P = 85^\circ$, and that $\angle SQT$ and $\angle PQR$ are vertical angles, find $m\angle R$. 2. **Use the Vertical Angle Theorem:** Vertical angles are congruent, so $$m\angle SQT = m\angle PQR$$ which means $$m\angle PQR = 30^\circ$$ 3. **Apply the Triangle Sum Theorem:** The sum of the interior angles of triangle $PQR$ is $$m\angle PQR + m\angle P + m\angle R = 180^\circ$$ Substitute the known values: $$30^\circ + 85^\circ + m\angle R = 180^\circ$$ 4. **Simplify the equation:** $$115^\circ + m\angle R = 180^\circ$$ 5. **Solve for $m\angle R$:** $$m\angle R = 180^\circ - 115^\circ$$ $$m\angle R = 65^\circ$$ **Final answer:** $$\boxed{65^\circ}$$