Final answer:
Option A is correct option. The congruency between line PN and line MQ depends on the specific lengths of PM and NQ; without additional information, no definitive relationship can be determined.
Step-by-step explanation:
If points M and N are on line PQ, and line PM is congruent to line NQ, we can assume that these segments are equal in length due to the definition of congruency. However, when considering the relationship between line PN and line MQ, their congruency depends on the specific lengths of PM and NQ rather than the fact that PM and NQ are congruent to each other. In other words, without additional information regarding the lengths of these segments or the positions of points M and N on line PQ, we cannot determine a definitive relationship between PN and MQ.
For instance, if points M and N divide the line PQ into three equal segments, then PN would be twice as long as PM or NQ, and MQ would also be twice as long. In this case, PN and MQ would indeed be congruent. However, if the segments PM and NQ are not half the length of PQ, PN and MQ may not be congruent.
Therefore, the correct option in this scenario is D) It depends on the specific lengths of PM and NQ.