If a line has a "beginning" but no "end" or if a line had no "beginning" but an "end" how could we say... ..?

To have a beginning and to be infinite are not contradictory properties. To be infinite does not mean to be all inclusive. The set of all odd numbers is an infinitely large set, but it does not mean that because it is infinite it must include all numbers. For example, such a set, though infinite, would not contain any even numbers.