A condition for intersection
The original statement supplies an intersection condition rather than merely defining parallel lines. Two right angles amount to 180 degrees; the condition specifies interior angles on one side and the direction in which the lines meet.
The ambient plane matters. Skew lines in space do not become parallel in this sense simply because they fail to intersect. The relevant sides and alternate-angle relationships require the shared planar setting.
Related results can require different premises
Proposition I.27 concludes that equal alternate angles imply parallel lines without using the fifth postulate. Assuming an intersection would conflict with a previously established exterior-angle result.
I.29 derives equal alternate angles and related facts from parallel lines, explicitly invoking the fifth postulate. Apparently converse statements need not have demonstrations resting on identical premises.
Understanding changes of assumption
Later geometers tried to derive the postulate from other assumptions or deny it and seek a contradiction. A conclusion unfamiliar from ordinary Euclidean diagrams is not necessarily a logical inconsistency.
Hyperbolic geometry supplies different parallel relationships. Models such as a disk represented within a Euclidean setting allow another rule system to be studied. What changes are assumptions and the model’s interpretation of objects, not a claim that one unchanged premise set proves opposite results.