Question

select the counter - example that shows that the conjecture is false. conjecture: a line s divides $overline{mn}$ into two line segments. so, the line s is the segment bisector of $overline{mn}$. if another line also divides $overline{mn}$, then line s is not the only segment bisector of $overline{mn}$. line s must intersect $overline{mn}$ at a right angle in order to be the segment bisector of $overline{mn}$. if line s intersects $overline{mn}$ anywhere besides the mid - point, it is not the segment bisector of $overline{mn}$. if line s intersects $overline{mn}$ at the mid - point, it is the perpendicular bisector of $overline{mn}$.

Explanation:

Step1: Recall segment - bisector definition

A segment bisector is a line, ray, or segment that divides a line segment into two equal parts. It must intersect the segment at its mid - point.

Step2: Analyze each option

• Option 1: Just because another line can bisect a segment doesn't mean the original line isn't a bisector. This is not a counter - example.
• Option 2: A segment bisector doesn't have to intersect at a right angle. A non - perpendicular line can also be a segment bisector if it passes through the mid - point. This is not a counter - example.
• Option 3: If a line intersects a segment anywhere besides the mid - point, it does not divide the segment into two equal parts. So it is not a segment bisector. This is a counter - example to the conjecture that just because a line divides a segment into two segments, it is a segment bisector.
• Option 4: Intersecting at the mid - point doesn't necessarily mean it is a perpendicular bisector. But this is not a counter - example to the given conjecture about a segment bisector.

Answer:

If line s intersects $\overline{MN}$ anywhere besides the midpoint, it is not the segment bisector of $\overline{MN}$.