Elementary Geometry

Introduction to Geometry
Geometry is a subject in mathematics that focuses on the study of shapes, sizes,
relative configurations, and spatial properties. Derived from the Greek word meaning
"earth measurement," geometry is one of the oldest sciences. It was first formally
organized by the Greek mathematician Euclid around 300 BC when he arranged 465
geometric propositions into 13 books, titled 'Elements'. This, however, was not the first
time geometry had been utilized. As a matter of fact, there exists evidence to believe that
geometry dates all the way back to 3,000 BC in ancient Mesopotamia, Egypt!

Geometry has been the subject of countless developments. As a result, many types
of geometry exist, including Euclidean geometry, non-Euclidean geometry, Riemannian
geometry, algebraic geometry, and symplectic geometry.

This discussion primarily focuses on the properties of lines, points, and angles. We
will also place emphasis on geometric measurements including lengths, areas, and
volumes of various shapes. By the end of this section it won't be hard to see that
geometry is all around us! The History of Geometry

Geometry's origins go back to approximately 3,000 BC in ancient Egypt. Ancient

Euclid's Elements
The next great advancement in geometry came from Euclid in 300 BC when he
wrote a text titled 'Elements.' In this text, Euclid presented an ideal axiomatic form (now
known as Euclidean geometry) in which propositions could be proven through a small set
of statements that are accepted as true. In fact, Euclid was able to derive a great portion
of planar geometry from just the first five postulates.
Below are some of the key concepts and terms you will need to know in order to begin
your study of geometry.

Points
In geometry, we use points to specify exact locations. They are generally denoted
by a number or letter. Because points specify a single, exact location, they are zero-
dimensional. In other words, points have no length, width, or height. It may be helpful to
think of a point as a miniscule "dot" on a piece of paper.

Lines
Lines in geometry may be thought of as a "straight" line that can be drawn on
paper with a pencil and ruler. However, instead of this line being bounded by the
dimensions of the paper, a line extends infinitely in both directions. A line is one-
dimensional, having length, but no width or height. Lines are uniquely determined by two
points. Thus, we denote the name of a line passing through the points A and B as, where
the two-headed arrow signifies that the line passes through those unique points and
extends infinitely in both directions.

Line Segments
Consider the task of drawing a "straight" line on a piece of paper (as we've done
when thinking about lines). What you've actually done is create a line segment. Because
our piece of paper has defined dimensions and we cannot draw a line infinitely in any
direction, we have constructed a segment that begins somewhere and ends somewhere.
We write the name of a line segment with endpoints A and B as. Note that the notation for
lines and line segments differ because a line segment has a defined length, where as a line
does not.

Rays
A ray is a "straight" line that begins at a certain point and extends infinitely in one
direction. A ray has one endpoint, which marks the position from where it begins. A ray
beginning at the point A that passes through point B is denoted as. This notation shows
that the ray begins at point A and extends infinitely in the direction of point B.

Endpoints
Endpoints mark the beginning or end of a line segment or ray. Line segments have
two endpoints, giving them defined lengths, whereas rays only have one endpoint, so the
length of a ray cannot be measured.

Midpoints
The midpoint of a line segment marks the point at which the segment is divided
into two equal segments. In other words, the lengths of the segments from either endpoint
to the midpoint are equal. Note that neither lines nor rays can have midpoints because
they extend infinitely in at least one direction. It would be impossible to find the middle
of a line or ray that never ends!

Intersection
When we have lines, line segments, or rays that meet, or cross at a certain point,
we call it an intersection point. In other words, those figures intersect somewhere.

Parallel
Two lines that will never intersect are called parallel lines. In the case of line
segments and rays, we must consider the lines that they lie in. In other words, we must
consider the case that the line segments or rays were actually lines that extend infinitely
in both directions. If the lines they lie on never intersect, they are called parallel. For
instance, the statement “is parallel to,".
Transversal
A transversal is a type of line that intersects at least two other lines. The lines that
a transversal crosses may or may not be parallel.

Planes
A plane can be thought of as a two-dimensional flat surface, having length and
width, but no height. A plane extends indefinitely on all sides and is composed of an
infinite number of points and lines. One way to think about a plane is as a sheet of paper
with infinite length and width.

Space
Space is the set of all possible points on an infinite number of planes. Thus, space
covers all three dimensions - length, width and height.