Answer to Question #299 in Geometry for Silvia Rosso

Question #299
Can a straight line, which intersects one of the parallel straight lines, do not intersect the other?
Expert's answer
It’s impossible if all three lines are in one plane. Let’s call the parallel lines l and 2, the line, which intersects one of the parallel lines – 3. Let 3 intersects 1 in the point called A. If 3 doesn’t intersect 2, 2 and 3 must be parallel. So we get that through the point A are drawn two lines, parallel to 2 – 1 and 3. According to the ninth Euclid's axiom through the point, which does not lie on the line, can be drawn no more than one parallel to it line. So, we get the contravention. But if 3 does not belong to the same plane as 1 and 2 belong, it can intersect 1 but do not intersect 2.

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