If we know at least one angle, we can always determine the remaining three around the same point of intersection by using the Vertical Angles Congruence Theorem or the Linear Pair Postulate. Below you see an example of how one angle is related to the other three.
Therefore, the only requirement is that we know at least one of the angles around each point of intersection. Thus, four examples that would allow us to determine if j∥k is if we knew:
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