Two collinear rays that do not intersect are called parallel rays. This will only occur for a short time, however, because they will intersect at some point.
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A vertex.
-- Two rays with a common endpoint form an angle. -- If they have a common endpoint, then they are collinear. So I guess, technically, the conditions described in the question aren't possible.
Perpendicular rays are two rays that intersect one another to form four right angles. Below is an image that shows two perpendicular rays.
Opposite Rays