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Your economy of expression works to obfuscate your meaning,

but I think you may be referring to a 'parallel' line.

Q: What is a line in the same plane that does not share a common point?

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Yes, it is possible.

point, line and plane

A ray

Angles that share a common end point and a common line segment but no common interior point (two angles that are next to each other)

When the line is inclined to the plane. That is, it is not in the plane nor is it parallel to it.

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Concurrent or intersecting lines.

Yes, it is possible.

Plane. A point has no dimension, a line has one dimension, and a plane has two dimensions.

point, line and plane

point * * * * * or, nothing (if the line is parallel to the plane).

Angles that share a common end point and a common line segment but no common interior point (two angles that are next to each other)

A ray

plane

Angles that share a common end point and a common line segment but no common interior point (two angles that are next to each other)

Two straight lines in the same plane can't possibly have more than one point in common, unless they are both the same line. If they're parallel, they have no common points. If they're not parallel, then have exactly one common point. If they're the same line, then every point on one line is also on the other one.

A plane intersects a line at a point, and i plane intersects another plane at a line.

When the line is inclined to the plane. That is, it is not in the plane nor is it parallel to it.