What is Euclidean Geometry?
Euclidean geometry is a type of geometry based on a collection of postulates formulated by the ancient Greek mathematician Euclid of Alexandria It is the oldest and most studied form of geometry, and encompasses plane geometry and solid geometry. It is a synthetic approach to geometry, which means that it deals with the joining of points and the drawing of lines, circles and other forms. Euclidean geometry was used extensively in architecture prior to the development of the more precise non-Euclidean geometry, which is used in modern engineering and other sciences.
Plane geometry deals with the properties of points and lines in a two dimensional Euclidean space. This type of geometry follows the rules of Euclidean postulates, such as the existence of infinitely many points and lines, the existence of perfect circles, the existence of precise angles, the existence of a line segment’s length is finite, and that the interior angles of a triangle sum up to 180 degrees. These postulates can be used to prove the propositions, which are then used to derive theorems.
Solid geometry deals with the properties of points and lines in three-dimensional Euclidean space. It follows the same postulates of plane geometry, but also introduces concepts such as surfaces, volumes and angles as well as the rules of congruence, similarity and Pythagorean theorem. Through these postulates, one can prove propositions, develop theorems and calculate area, volume and angles.
Following the postulates of Euclidean geometry, the five best examples are as follows:
1. The Pythagorean theorem, which states that in a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.
2. The theorem of the inscribed angles, which states that in a triangle, the angle between two sides is equal to half the angle between the other sides.
3. The theorem of the exterior angles, which states that in a triangle, the exterior angle is equal to the sum of the two interior angles.
4. The theorem of the sum of angles of a triangle, which states that in a triangle, the sum of its angles is equal to 180 degrees
5. The theorem of the base angles, which states that in a triangle, the base angles are equal.
These theorems and postulates provide a basis for further exploration of geometry, and can be used to discover new relationships between shapes, areas, lengths and angles. They are some of the most basic concepts in Euclidean geometry, and are fundamental to understanding its principles. In this way, Euclidean geometry is an important part of mathematics, and its principles are essential to the study of other branches of mathematics such as calculus and other forms of geometry.