Plane geometry refers to the geometry constructed according to Euclid's Elements. It is also called Euclidean geometry. Plane geometry studies the geometric structure and metric properties (area, length, angle) of straight lines and quadratic curves (i.e. conic curves, i.e. ellipses, hyperbolas and parabolas) on a plane. Plane geometry uses an axiomatic approach and has important significance in the history of mathematical thought.

Euclidean geometry is sometimes referred to as geometry on the plane, or plane geometry. This article focuses on plane geometry. Euclidean geometry in three dimensions is usually called solid geometry. For higher dimensions, see Euclidean space. In mathematics, Euclidean geometry is the common geometry in the plane and three dimensions, based on the point-line-plane hypothesis. Mathematicians also use this term to refer to higher-dimensional geometry with similar properties.

Among them, Postulate 5 is also called the Parallel Axiom, which is more complicated to describe. This postulate derives the theorem that "the sum of the interior angles of a triangle is equal to 180 degrees". In the era of F. Gauss (1777-1855), Postulate 5 was questioned. Russian mathematician Nikolay Ivanovitch Lobachevski and Hungarian mathematician Bolyai explained that the fifth postulate is only a possible choice of the axiom system, not an inevitable geometric truth, that is, "the sum of the interior angles of a triangle is not necessarily equal to 180 degrees", thus discovering non-Euclidean geometry, namely "non-Euclidean geometry".