Not necessarily.
Imagine yourself inside a cuboid room.
Consider the following three lines:
(A) The horizontal line joining the far wall and the floor.
(B) The horizontal line joining the wall on your left and the ceiling.
and
(C) The vertical line joining the far wall and the wall on your left.
The line C may be considered a transversal to the other two. These are both parallel but they are not coplanar. Their planes are both horizontal but Line A is in a low plane while B is in a high plane.
a transversal
Usually, a transversal is a line that intersects two (or more) parallel lines. In that case the lines and the transversal are coplanar. However, a transversal does not have to intersect parallel lines. And in that case, the lines need not be coplanar. Here's one way to visualise the latter situation. Stand in a cuboid room. Line one = the edge joining the wall opposite you to the ceiling. Line two = the edge joining the wall on your right to the floor. Transvesal = the edge joining the opposite wall to the wall on your right. The transversal meets both the two lines but lines 1 and 2 are not coplanar.
transversal
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corresponding angles
transversal
a transversal
A transversal.
They are always coplanar in Euclidean geometry.
Alternate Interior Angles
Transversal
Usually, a transversal is a line that intersects two (or more) parallel lines. In that case the lines and the transversal are coplanar. However, a transversal does not have to intersect parallel lines. And in that case, the lines need not be coplanar. Here's one way to visualise the latter situation. Stand in a cuboid room. Line one = the edge joining the wall opposite you to the ceiling. Line two = the edge joining the wall on your right to the floor. Transvesal = the edge joining the opposite wall to the wall on your right. The transversal meets both the two lines but lines 1 and 2 are not coplanar.
transversal
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corresponding angles
Normally, yes. A transversal contemplates crossing two (normally parallel) lines in conversations about two dimensional space and the relationship of certain angles. If you are talking about three dimensions, all bets are off. Two skewed lines in three dimensional space could would have a line that connects them but none of them would be coplanar.
Two pairs of alternate opposite angles