Final answer:
To prove PQ = LM, it's established that Q and M are the midpoints of congruent line segments PR and LN, respectively. Thus, PQ and LM are each half the length of PR and LN, making them equal.
Step-by-step explanation:
The solution involves understanding the properties of congruent line segments and midpoints. Given that line segment PR is congruent to line segment LN, and that Q is the midpoint of PR and M is the midpoint of LN, we need to show that PQ is equal to LM.
Since Q and M are the midpoints of PR and LN respectively, then by the definition of a midpoint, PQ is half the length of PR, and LM is half the length of LN. Because PR is congruent to LN, they have the same length. Therefore, half of PR (which is PQ) must be equal to half of LN (which is LM).
So it's proven that PQ equals LM because both are halves of congruent line segments, PR and LN, respectively.