Question

1. what additional information would you need to conclude that $overleftrightarrow{jh}$ is the perpendicular bisector of $overline{kl}$?

$overline{kj}$ is perpendicular to $overleftrightarrow{jh}$
$overline{kl}$ is perpendicular to $overleftrightarrow{jh}$
$overline{hi}congoverline{ij}$
$overline{jh}congoverline{kl}$

Explanation:

Step1: Recall definition of perpendicular bisector

A perpendicular bisector of a line - segment is perpendicular to the line - segment and divides it into two equal parts.

Step2: Analyze each option

• Option 1: $\overline{KJ}$ is perpendicular to $\overleftrightarrow{JH}$ only tells about the perpendicularity of $\overline{KJ}$ and $\overleftrightarrow{JH}$, not about $\overline{KL}$ and $\overleftrightarrow{JH}$.
• Option 2: If $\overline{KL}$ is perpendicular to $\overleftrightarrow{JH}$, and from the figure, it seems that $I$ is the mid - point of $\overline{KL}$ (the red marks suggest equal segments), then $\overleftrightarrow{JH}$ is the perpendicular bisector of $\overline{KL}$. This option satisfies the conditions for a perpendicular bisector.
• Option 3: $\overline{HI}\cong\overline{IJ}$ gives information about the segments on $\overleftrightarrow{JH}$ and not about the relationship between $\overleftrightarrow{JH}$ and $\overline{KL}$.
• Option 4: $\overline{JH}\cong\overline{KL}$ gives information about the lengths of two different line - segments and not about the perpendicular bisector relationship.

Answer:

$\overline{KL}$ is perpendicular to $\overleftrightarrow{JH}$