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.
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Lines of latitude.
Yes. That's a big part of the reason why the lines are often called "parallels".
parallel lines. they run side by side but never intersect
parallel lines.....
Parallels are lines that run the same direction and are the always the same distance apart. On a map (and the Earth) the lines of latitude (measurement North and South of the equator) are parallel.