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The only line that runs parallel through the equator is THE EQUATOR. [The Tropic of Cancer and the Tropic of Capricorn are parallel to the equator, but are north and south of it (respectively) at 23.5 degrees. So they do not run through the equator.] The lines of longitude all run through the equator, but they are not parallel to each other since they all meet up at both the North and South Poles.
parallel lines. they run side by side but never intersect
parallel lines.....
Parallel lines run in the same direction. If they are intersected by one line and that line is perpendicular to both, then it proves that the lines run parallel to each other.
Lines of latitude are horizontal and run parallel to the poles