[general_re]

Which should hardly be a surprise, given the nature of the triangle we are describing, don't you think? ;)

IOW, the law of cosines still holds true, even in such a case - it is a law, after all. So to define an angle as "two lines diverging from a common point" is too restrictive - defining it in such a manner implicitly assumes that there is a length involved, when the formal relationship between the sides and angles of a triangle requires no such thing.

[AndrewC]

P.S. If you want an illustration go here and set all the sides to zero and click on Angle C. Now put 1,9,1 as the sides and click on Angle C. Same answer. 1,9,1 is not a triangle and neither is 0,0,0.

[BMCDA]

A length is required.
A line is defined by two points that don't have the same coordinates and for an angle you need at least three points which don't share the same coordinates. These three points form the two diverging lines that comprise an angle (if two points have the same coordinates you have an angle of "0").
So if these points must have different coordinates in order to make sense, you automatically have lenghts that are strictly greater than zero.