1. **State the problem:** We have a parallelogram RSTU with diagonals intersecting at point M. Given lengths are $RS=9$, $ST=20$, and $RM=11$. We need to determine which segment is longer: $STM$ or $RM$. 2. **Recall properties of parallelograms:** In a parallelogram, the diagonals bisect each other. This means point $M$ is the midpoint of both diagonals $RT$ and $SU$. 3. **Analyze the diagonal $RT$:** Since $M$ is the midpoint of $RT$, the segments $RM$ and $MT$ are equal. Given $RM=11$, it follows that $MT=11$. 4. **Consider segment $STM$:** The segment $STM$ is composed of points $S$, $T$, and $M$. Since $ST=20$ and $TM=11$ (because $M$ lies on diagonal $RT$ and $T$ is a vertex), the length of $STM$ is the sum of $ST$ and $TM$. 5. **Calculate length of $STM$:** $$STM = ST + TM = 20 + 11 = 31$$ 6. **Compare lengths:** - $RM = 11$ - $STM = 31$ Therefore, $STM$ is longer than $RM$. **Final answer:** Segment $STM$ is longer than segment $RM$ because $STM=31$ and $RM=11$.