plane geometry refers to geometry constructed according to euclid's elements of geometry. 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 sections, i.e. ellipses, hyperbolas and parabolas) on the plane. plane geometry adopts the axiomatic method, which is of great significance in the history of mathematical thought.

euclidean geometry sometimes refers to geometry on the plane, that is, plane geometry. this article mainly describes plane geometry. euclidean geometry in three dimensions is often called solid geometry. for high-dimensional situations, please refer to euclidean space. mathematically, euclidean geometry is common geometry in planes and three-dimensional spaces, based on the point-line-plane hypothesis. mathematicians also use the term to refer to higher-dimensional geometries with similar properties.

among them, postulate five is also called the parallel axiom, and the description is relatively complicated. this postulate derives the theorem that "the sum of the interior angles of a triangle is equal to one hundred and eighty degrees." in the era of f. gauss (1777-1855), postulate 5 was highly questioned. russian mathematician nikolay ivanovitch lobachevski and the hungarian bolyai clarified that the fifth postulate is only a possible choice of the axiom system and is not an inevitable geometric truth, that is, "the sum of the interior angles of a triangle is not necessarily equal to one hundred and eighty degrees", thereby discovering non-euclidean reid's geometry is "non-euclidean geometry".