What is the definition of parallel lines?

As a geometry enthusiast, I love this question! 📐 In a Euclidean plane, parallel lines are two distinct lines that are always the same distance apart (equidistant) and will never intersect, no matter how far you extend them. The formal definition usually involves their slopes: * Two non-vertical lines are parallel if and only if they have the exact same slope ($m_1 = m_2$). * Vertical lines (which have an undefined slope) are parallel to all other vertical lines. Think of the tracks of a straight train railway — they run alongside each other forever without ever crossing. That's the perfect real-world example!

Parallel lines are two or more straight lines in a plane that never intersect, no matter how far extended. They maintain a constant distance and have the same slope in Euclidean geometry. Example: railway tracks are ideally parallel. Symbolically, if line l is parallel to line m, we write l ∥ m.

Simply put, parallel lines are lines in a plane that never meet. Never. It's like they're in a perpetual, straight-line standoff. The key is that they must be coplanar (exist on the same flat surface, or plane). If they are in 3D space and don't intersect but aren't parallel, we call them skew lines, so don't get those confused! Parallel is all about that constant, unwavering distance between them.

To make it even more formal, the existence of parallel lines is guaranteed by the Parallel Postulate (or Euclid's Fifth Postulate), which basically states that for any given line and a point not on that line, there is exactly one line through the point that is parallel to the given line. This is the foundation of traditional Euclidean geometry! So, the definition is not just descriptive, but foundational to the whole system.