Euclidean Geometry Fundamentals

Study Notes

Definition and Axioms

Euclidean geometry is a system of geometry that is based on the works of the ancient Greek mathematician Euclid
It is a mathematical system that deals with the study of shapes, sizes, and positions of objects
Euclidean geometry is based on five axioms (self-evident truths) and five postulates (assumptions)

Axioms

It is possible to draw a straight line from any point to any other point
It is possible to extend a finite straight line continuously in both directions
It is possible to describe a circle with any center and any radius
All right angles are equal to one another
Things that are equal to the same thing are also equal to each other

Postulates

It is possible to draw a straight line from any point to any other point
It is possible to extend a finite straight line continuously in both directions
It is possible to describe a circle with any center and any radius
That all right angles are equal to one another
That, if a straight line falling on two straight lines makes the interior angles on one side less than two right angles, then the two straight lines, if produced far enough, meet on that side on which the angles are less than two right angles

Key Concepts

Points: A location in space, represented by a set of coordinates
Lines: A set of points that extend infinitely in two directions
Rays: A line that extends from a point to infinity in one direction
Angles: The measure of the amount of rotation between two lines or planes
Triangles: A polygon with three sides and three angles
Congruent: Two shapes that have the same size and shape
Similar: Two shapes that have the same shape but not necessarily the same size

Theorems

Pythagorean Theorem: In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides
Thales' Theorem: The angle inscribed in a semicircle is a right angle
Angle-Sum Theorem: The sum of the interior angles of a triangle is equal to two right angles

Applications

Architecture: Used to design and construct buildings, bridges, and other structures
Engineering: Used to design and calculate the stresses and strains on buildings, bridges, and other structures
Art: Used to create perspective and proportion in art
Physics: Used to describe the motion of objects and the geometry of space and time

Description

Learn the basics of Euclidean geometry, including axioms, postulates, key concepts, and theorems. Understand the principles of points, lines, angles, and triangles, and explore its applications in architecture, engineering, art, and physics.