Geometry

C

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.

D

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.

---

---

III

Euclidean Geometry

Perhaps the most familiar and intuitive geometry is called Euclidean geometry. Euclidean geometry describes most aspects of the everyday world and was named after Euclid, the ancient Greek mathematician who developed it. While the postulates of Euclidean geometry do seem plausible when applied to physical space in our universe, there is evidence that Euclidean geometry is not the perfect system for describing space.

Two-dimensional Euclidean geometry is often called plane geometry; three-dimensional Euclidean geometry is frequently referred to as solid geometry. Plane geometry deals with figures that lie wholly in one plane. A plane may be measured in terms of two dimensions: length and width. Solid geometry deals with figures that have three dimensions: length, width, and height.

Conic sections, a commonly studied topic of geometry, are two-dimensional curves created by slicing a plane through a three-dimensional hollow cone.

A

Euclid’s Postulates

Euclid, who lived about 300 bc, realized that only a small number of postulates underlay the various geometric theorems known at the time. He determined that these theorems could be deduced from just five postulates.

1. A straight line may be drawn through any two given points.

2. A straight line may be drawn infinitely or be limited at any point.

3. A circle may be drawn using any given point as the center, and with any given radius (the distance from the center to any point on the circle).

4. All right angles are congruent. (A right angle is an angle that measures 90°. Two geometric figures are congruent if they can be moved or rotated so that they exactly overlap.)

5. Given a straight line and a point that does not lie on the line, one and only one straight line may be drawn that is parallel to the first line and passes through the point.

These five postulates can be used in combination with various defined terms to prove the properties of two- and three-dimensional figures, such as areas and circumferences. These properties can in turn be used to prove more complex geometric theorems.

B

Two-Dimensional Euclidean Figures

Figures commonly encountered in two-dimensional geometry include circles, polygons, triangles, and quadrilaterals. Triangles are actually three-sided polygons; quadrilaterals are polygons with four sides.

	More from Encarta
	•  Important Notice: MSN Encarta to be discontinued. Learn more. •  Presidential Myths Quiz •  Coffee break: Recharge your brain