Question

construct a segment congruent to \\(\overline{kl}\\) on \\(\overrightarrow{mn}\\) with one endpoint at m.

Explanation:

Step1: Measure length of $\overline{KL}$

Use a compass to set the distance between its two ends equal to the length of segment $\overline{KL}$ (by placing the compass point on $K$ and the pencil on $L$).

Step2: Draw arc on $\overrightarrow{MN}$

With the compass still set to the length of $\overline{KL}$, place the compass point on $M$ (the given endpoint on $\overrightarrow{MN}$) and draw an arc that intersects $\overrightarrow{MN}$. Let the intersection point be $P$.

Step3: Confirm congruent segment

The segment $\overline{MP}$ is now constructed such that $MP = KL$, so $\overline{MP}\cong\overline{KL}$ with one endpoint at $M$ on $\overrightarrow{MN}$.

Answer:

The constructed segment is $\overline{MP}$ (where $P$ is the intersection point from the arc drawn with compass centered at $M$ with radius equal to $KL$ length) congruent to $\overline{KL}$ on $\overrightarrow{MN}$ with endpoint at $M$.