Geometry is a branch of mathematics that deals with shapes
and sizes. Geometry may be thought of as the science of space. Just as
arithmetic deals with experiences that involve counting, so geometry describes
and relates experiences that involve space. Basic geometry allows us to
determine properties such as the areas and perimeters of two-dimensional shapes
and the surface areas and volumes of three-dimensional shapes. People use
formulas derived from geometry in everyday life for tasks such as figuring how much
paint they will need to cover the walls of a house or calculating the amount of
water a fish tank holds.
Geometry combines simple conceptual building blocks to
construct complex logical structures. These building blocks include undefined
terms, defined terms, and postulates. Combining these components creates chains
of reasoning that support conclusions called theorems.
Undefined terms
Some concepts central to geometry are not defined in terms
of simpler concepts. The most familiar of these undefined terms are point,
line, and plane.
These fundamental concepts arose from everyday experiences. Thus, the experience of where an object is leads to the idea of an exact, fixed location. This is the intuitive idea to which the term point refers. Many physical objects suggest the idea of a point. Examples include the corner of a block, the tip of a pencil, or a dot on a sheet of paper. Such things are called models or representations or pictures of points, although they show only approximately the idea in mind. Similarly, a row of points suggested by a tightly stretched string, the edge of a desk, or a flagpole, extended infinitely in both directions, is called a line. The word plane describes a flat surface—such as a floor, desktop, or chalkboard—but it is imagined as extending infinitely in all directions. This means that a plane has no edges just as a line has no ends.
Other undefined terms describe relations among points,
lines, and planes, such as the relation described by the phrase “a point that
lies on a line.”
Defined Terms
Undefined terms can be combined to define other terms.
Noncollinear points, for example, are points that do not lie on the same line.
A line segment is the portion of a line that includes two particular points and
all points that lie between them, while a ray is the portion of a line that
includes a particular point, called the end point, and all points extending
infinitely to one side of the end point.
Defined terms can be combined with each other and with
undefined terms to define still more terms. An angle, for example, is a
combination of two different rays or line segments that share a single end
point. Similarly, a triangle is composed of three noncollinear points and the
line segments that lie between them.
Postulates
Postulates, or axioms, are unproven but universally accepted
assumptions, such as “there is one and only one line that passes through two
distinct points.” A system consisting of a set of noncontradictory postulates
concerning the undefined terms point, line, and plane, together with the
theorems deduced from these postulates, is called a geometry. Different sets of
postulates determine whole different systems of geometry.
If the postulates selected are suggested by experience with
physical space, then it is reasonable to expect that the conclusions will also
correspond closely to experiences related to space. However, since any set of
postulates must be selected on the basis of incomplete and approximate
observation, they quite possibly apply only approximately to actual space.
Thus, it is no surprise if any particular geometry should turn out to be
inapplicable, or only approximately applicable, to problems in actual space.
Theorems
Theorems are logically deduced from postulates. This process
of deduction is called a proof. Each step of a proof must be justified by one
of the postulates or by a theorem that has already been proved. One simple
theorem, for example, asserts that a line that is parallel to one of a pair of
parallel lines is parallel to both lines. Parallel lines are lines that are
equally far apart from each other along their entire lengths.
In proving a theorem in geometry, we deduce a conclusion
from a set of assumptions. (Encarta Encyclopedia)