Question

use the compass and straightedge to construct a segment congruent to $overline{mn}$ with endpoint $p$.

Explanation:

Step1: Set compass width

Open the compass so that its ends align with points $M$ and $N$. This sets the compass width to the length of segment $\overline{MN}$.

Step2: Draw an arc

With the compass point on $P$, draw an arc.

Step3: Mark the new point

Using the straight - edge, choose a point $Q$ on the arc. The segment $\overline{PQ}$ is congruent to $\overline{MN}$.

Answer:

Constructed segment $\overline{PQ}$ (where $Q$ is a point on the arc drawn from $P$ with compass width equal to $MN$) is congruent to $\overline{MN}$.