To find the lengths of the line segments $\overline{LM}$ and $\overline{PQ}$, you need the coordinates of the points $L$, $M$, $P$, and $Q$. Once you have the coordinates, you can use the distance formula to calculate the lengths of both segments.

The distance formula for two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Steps:

1. Find the length of $\overline{LM}$: Using the coordinates of $L(x_1, y_1)$ and $M(x_2, y_2)$, apply the distance formula.

2. Find the length of $\overline{PQ}$: Using the coordinates of $P(x_3, y_3)$ and $Q(x_4, y_4)$, apply the same distance formula.

3. Determine congruency: Compare the lengths of $\overline{LM}$ and $\overline{PQ}$. If the lengths are equal, the segments are congruent; otherwise, they are not.

Could you provide the coordinates of the points, or would you like me to continue with hypothetical values?

If you have any questions or need further details, feel free to ask! Here are some related questions to explore:

1. What does it mean for two segments to be congruent?
2. How is the distance formula derived from the Pythagorean theorem?
3. Can line segments be congruent if their coordinates are different?
4. What are some practical uses of congruent segments in geometry?
5. How do you verify the congruence of shapes using side lengths?

Tip: The distance formula is essentially the Pythagorean theorem applied in a coordinate plane!