4: Basic Concepts of Euclidean Geometry

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Learning Objectives

Develop students':

ability to visualize problems
familiarity and facility with a wide range of geometry facts and problem-solving techniques
understanding of the logical structure of geometry-axioms, conjectures, theorems and counterexamples
visual reasoning

In Greek, "geo" means earth, and "metron” means measure. Egyptians were among the first people to use geometry to survey the land. The study of geometry was carried on by the Greeks Thales, Pythagoras (550 BC), Plato, and Euclid ( father of geometry, 300 BC). They not only asked “how” and “what” but also asked, “why.” At the foundations of any theory, there are truths, which are taken for granted and can't be proved or disproved. These are called axioms. The first axiomatic system was developed by Euclid in his books called "Elements". Research in teaching and learning of geometry has given strong support to the van Hiele theory. This theory was developed in the late 1950s by two Netherlands mathematics teachers. they observed that in learning geometry, students seem to progress through the following sequence: recognition, analysis, relationship, deduction (the study of geometry as a mathematical system), and axiomatic.

4.1: Euclidean geometry
Euclidean geometry, sometimes called parabolic geometry, is a geometry that follows a set of propositions that are based on Euclid's five postulates. There are two types of Euclidean geometry: plane geometry, which is two-dimensional Euclidean geometry, and solid geometry, which is three-dimensional Euclidean geometry.
4.2: 2-D Geometry
A polygon is a closed, 2-dimensional shape, with edges(sides) are straight lines. The word “polygon” is derived from Greek for “many angles”. The names of the polygons are taken from the Greek number prefixes followed by –gon, with only a couple exceptions.
4.3: 3-D Geometry
Polyhedra are simple 3D closed surfaces that are composed of polygonal regions.
4.4: Transformations
A translation is a transformation that moves a figure (without altering dimensions) to a new position.
4.5: Symmetry
In 2D geometry, a figure is symmetrical if an operation can be done to it that leaves the figure occupying an identical physical space. This can be accomplished in two ways.
4.6: Summary
Geometry has an algebra all its own. Operations (translations) can be done to geometric figures.
4.E: Basic Concepts of Euclidean Geometry (Exercises)

Thumbnail: This image illustrates in 3D a stereographic projection from the north pole onto a plane below the sphere. (CC BY-SA 4.0; Mark.Howison).

Contributor

Pamini Thangarajah (Mount Royal University, Calgary, Alberta, Canada)