Bolyai–Lobachevsky Theorem

The Bolyai–Lobachevsky Theorem states that in hyperbolic geometry—which differs from the familiar Euclidean geometry—the angles of a triangle add up to less than 180 degrees. As you construct larger triangles, the difference from 180 degrees grows, allowing for many unique properties such as multiple parallel lines through a point outside a given line. Essentially, this theorem confirms that hyperbolic space behaves differently from flat space, leading to rich and consistent geometric structures where the usual rules about angles and lines are altered.