Question

which of the following statements must be true based on the diagram below? select all that apply. (diagram is not to scale.)

answer
attempt 1 out of 2
$overline{mn}$ is a segment bisector.
$overline{mn}$ is a perpendicular bisector.
$n$ is the vertex of two angles that are congruent to one another.
$m$ is the vertex of a right angle.
$n$ is the midpoint of a segment in the diagram.
none of the above.

Explanation:

1. For $\overline{MN}$ as a segment bisector: A segment bisector divides a segment into two equal parts. $\overline{MN}$ intersects $\overline{LJ}$, but we only know $M$ is a point on $\overline{LJ}$ with markings on either side, but $\overline{MN}$ does not divide $\overline{LJ}$ (it just meets it at $M$) and does not divide another segment into two equal parts, so this is false.
2. For $\overline{MN}$ as a perpendicular bisector: A perpendicular bisector must be perpendicular to a segment and bisect it. $\overline{MN}$ is perpendicular to $\overline{LJ}$, but it does not bisect $\overline{LJ}$ (the markings show $M$ is not the midpoint of $\overline{LJ}$), so this is false.
3. For $N$ as vertex of two congruent angles: $N$ is on $\overline{LK}$, which has $N$ as its midpoint, but there is no indication that angles at $N$ are congruent, so this is false.
4. For $M$ as vertex of a right angle: The right angle symbol at $M$ confirms $\angle LMN$ (and $\angle JMN$) are right angles, so this is true.
5. For $N$ as midpoint of a segment: The congruent markings on $\overline{LN}$ and $\overline{NK}$ show $N$ is the midpoint of $\overline{LK}$, so this is true.

Answer:

• $M$ is the vertex of a right angle.
• $N$ is the midpoint of a segment in the diagram.