Exercise 32 Page 331

Let's start with examining the given diagram. We will name some of the points for the purposes of the solution.

We can see that angles $\angle MNL$ and $\angle LNK$ are congruent and have the same measure. Thus, $\overrightarrow{NL}$ is the bisector of $\angle MNK.$ Let's now recall what the Angle Bisector Theorem states. Point $L$ lies on the angle bisector $\overrightarrow{NL}.$ According to this theorem, it must be equidistant from the sides of the angle $\angle MNK$ — $\overrightarrow{NK}$ and $\overrightarrow{NM}.$ From the diagram, it might seem like $\overline{ML}$ and $\overline{LK}$ are the distances from $L$ to $\overrightarrow{NK}$ and $\overrightarrow{NM}.$ Then we would conclude that they are the same and $x$ equals $5.$ However, is it really so?

Let's remember that the distance from a point to a line/arrow is the length of the line segment, which joins the point to the line and is perpendicular to the line, as illustrated below.

We do not know if the segments $\overline{ML}$ and $\overline{LK}$ are perpendicular to $\overrightarrow{NK}$ and $\overrightarrow{NM}$ or not (and from what we see on the above diagram, it's possible that they are not). Thus, these may not be the distances from $L$ to the sides of the angle, and we cannot use the statement of the theorem. Therefore, there is not enough information to find $x.$