"Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid."
This topic introduces the basic building blocks of geometry, including points (which have no dimensions), lines (which have infinite length and no width), and planes (which have infinite length and width but no thickness).
Point: A point is a location that has no size or shape, represented by a dot.
Line: A line is a straight path that extends infinitely in both directions, represented by a series of points connected by a straight line.
Plane: A plane is a flat, two-dimensional surface that extends infinitely in all directions, represented by a bounding rectangle.
Intersection: The point where two or more lines, segments, or planes meet.
Parallel lines: Lines that have the same slope and never intersect, represented by two parallel lines.
Perpendicular lines: Lines that intersect at right angles, represented by a small square at the intersection.
Ray: A ray is a segment that starts at a point and extends infinitely in one direction, represented by an arrow pointing in the direction of the line.
Angle: An angle is a measure of the amount of turn between two lines that intersect at a point.
Congruent: Two shapes are congruent if they have the same shape and size, represented by an equal sign above the shapes.
Midpoint: The point that divides a segment into two equal parts.
Bisector: A line or plane that cuts a segment or angle into two equal parts.
Collinear: Three or more points that lie on the same line.
Coplanar: Points or lines that lie in the same plane.
Distance: The length of a path between two points, represented by a straight line between the points.
Coordinate plane: A plane divided into a grid of horizontal and vertical lines, used to locate or plot points.
Slope: The steepness of a line, represented by the rise over the run.
Transversal: A line that intersects two or more other lines or planes.
Vertical angles: Angles that are opposite each other and have equal measure, represented by a small square at the intersection.
Symmetry: A shape or object that can be divided into two equal parts that mirror each other.
Real Point: A point with definite position in space.
Ideal Point: A point at infinity, which has no definite position in space.
Locus Point: A point that satisfies a given condition, such as the intersection of two lines.
Midpoint: A point that divides a line segment into two equal parts.
Endpoint: A point that marks the beginning or the end of a line segment, ray or line.
Real Line: An infinitely long line with definite position in space.
Ideal Line: A line at infinity, which has no definite position in space.
Parallel Lines: Two lines that never intersect, no matter how far they are extended.
Intersecting Lines: Lines that meet or intersect each other.
Perpendicular Line: A line that intersects another line at a right angle.
Transversal Line: A line that intersects two or more parallel lines.
Skew Lines: Two lines that do not intersect and are not parallel.
Real Plane: An infinitely large flat surface with definite position in space.
Ideal Plane: A plane at infinity, which has no definite position in space.
Horizontal Plane: A plane that is perpendicular to the vertical axis, i.e., it is parallel to the ground.
Vertical Plane: A plane that is perpendicular to the horizontal axis, i.e., it is perpendicular to the ground.
Angled Plane: A plane that is not perpendicular to any of the axes.
Parallel Planes: Two planes that never intersect, no matter how far they are extended.
Intersecting Planes: Planes that meet or intersect each other.
Perpendicular Planes: Two planes that intersect at a right angle.
"Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these."
"Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems."
"The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions."
"Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language."
"Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that theorems proved from them were deemed absolutely true, and thus no other sorts of geometry were possible."
"Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century."
"An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances."
"Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects."
"This is in contrast to analytic geometry, introduced almost 2,000 years later by René Descartes, which uses coordinates to express geometric properties by means of algebraic formulas."
"the first ones having been discovered in the early 19th century."
"still taught in secondary school (high school) as the first axiomatic system"
"introduced almost 2,000 years later by René Descartes"
"he first to organize these propositions into a logical system"
"Euclidean space is a good approximation for it only over short distances"
"his textbook on geometry, Elements."
"assuming a small set of intuitively appealing axioms (postulates)"
"plane geometry, still taught in secondary school (high school) as the first axiomatic system"
"propositions about geometric objects"
"explained in geometrical language."