The problem in the image involves a triangle $\triangle MON$ with lines $s$ and $t$ acting as the perpendicular bisectors of sides $NO$ and $MO$, respectively. This geometric setup leads us to a few key observations about the relationships between the points and angles.

Given Information:

• Lines $s$ and $t$ are perpendicular bisectors.
• $s$ bisects side $NO$, meaning that it cuts $NO$ into two equal parts at point $P$.
• $t$ bisects side $MO$, cutting it into two equal parts at point $R$.
• Point $Q$ is the intersection of $s$ and $t$, and is the circumcenter of $\triangle MON$, meaning that it is equidistant from the vertices $M$, $O$, and $N$.

Statement Analysis:

1. $QR = RN$: Since $Q$ is the circumcenter, it is equidistant from all the vertices of the triangle, including $N$. Therefore, $QR$ cannot equal $RN$, as $Q$ is equidistant from $R$ and $N$, but these distances differ. (False)

2. $MP = PO$: Since $s$ is the perpendicular bisector of side $NO$, point $P$ is the midpoint of $NO$, making $MP \neq PO$ in this context. (False)

3. $\angle QNR \cong \angle QNM$: This statement suggests that the two angles formed by point $Q$ are congruent. Since point $Q$ is the circumcenter, the angles between these points are equal. (True)

4. $\angle QOR \cong \angle QOP$: Since $s$ and $t$ are perpendicular bisectors, and the angles formed around the circumcenter are symmetrical, this is a plausible statement. (True)

Conclusion:

The correct answer, based on the information, is $\angle QNR \cong \angle QNM$, as this geometric relationship must always hold true when $Q$ is the circumcenter of the triangle.

Would you like further clarification or details on how the other answers were determined?

Here are five follow-up questions to expand on this topic:

1. What properties do circumcenters have in different types of triangles (e.g., acute, obtuse)?
2. How are the perpendicular bisectors of a triangle related to its circumcenter?
3. Can a circumcenter lie outside a triangle? If so, under what conditions?
4. What is the relationship between the circumcenter and the orthocenter of a triangle?
5. How do the perpendicular bisectors help in constructing the circumcircle?

Tip: Remember, the circumcenter is always equidistant from all the vertices of the triangle, making it a key point in circumscribing circles.