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There is no specific name - other than what you used: "a pair of parallel line segments".
A circle.
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First, lines and rays cannot be congruent because they do not have both end points defined. Not having an end point define means we cannot talk about their length so they can't be congruent. Line segments, however, can be congruent. Congruent line segments NEED NOT be parallel.Furthermore, if two line segments are parallel, that does not mean they are congruent since parallel line segments may not have the same length.Congruent line segments would be ones that are the same length.