In projective geometry, any pair of lines always intersects at some point, but parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity.

Explanation of How Parallel Lines Meet at Infinity:
Parallel lines are lines that never intersect, no matter how far they are extended. However, in projective geometry, parallel lines are considered to meet at a point called the "point at infinity." This concept helps explain how parallel lines intersect in a way that is different from traditional Euclidean geometry.

Projective Geometry:
In projective geometry, points at infinity are added to the Euclidean plane to create a more complete and unified system. These points at infinity are considered to be at an infinite distance away, allowing parallel lines to meet at a common point.

Intersection at the Point at Infinity:
When parallel lines are extended to infinity in projective geometry, they appear to converge and meet at a single point known as the point at infinity. This point serves as a way to unify the parallel lines and provide a sense of closure in the system.

Visual Representation:
In a visual representation, parallel lines on a plane can be extended to the edges of the plane and beyond, eventually converging at the point at infinity. This concept helps in understanding how parallel lines can meet in a different way than we traditionally think.

Significance in Geometry:
The concept of parallel lines meeting at infinity is significant in projective geometry as it allows for a more complete and unified understanding of geometric concepts. It also helps in solving problems that involve parallel lines and their properties.
In conclusion, parallel lines meet at infinity in projective geometry by extending them to a point at an infinite distance away, known as the point at infinity. This concept enhances our understanding of geometric relationships and provides a more comprehensive framework for solving geometric problems.