What undefined term is described as no thickness but its length extends infinitely in both directions?

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           It wasn't until after the discovery of non-Euclidean geometry that mathematicians began examining the foundations of Euclidean geometry and formulating precise sets of axioms for it. The problem was to erect the entire structure of Euclidean geometry upon the simplest foundation possible; i.e. to choose a minimum number of undefined elements and relations and a set of axioms concerning them, with the property that all of the Euclidean geometry can be logically deduced form these without further appeal to intuition. Hilbert's approach does address Euclid's lack of attention to the notion of undefined terms and the concepts of incidence, betweenness and congruence. An example of Hilbert's precision and detail was to distinguish between a line and a line segment, as Euclid did not. This topic details Hilbert's undefined terms and preliminary definitions which can be used to provide the basis for traditional Euclidean geometry. A famous quote from Hilbert: "One must be able to say at all times-instead of points, lines, and planes---tables, chairs, and beer mugs."            A mutual understanding of the following terms is assumed: point, line, lie on, between, congruent, set, element of, intersection, and union. For example, two lines intersect means there is one point that lies on both of them; or said differently, two lines are incident (have a point in common).            In geometry, definitions are formed using known words or terms to describe a new word. There are three words in geometry that are not formally defined. These three undefined terms are point, line and plane.

POINT (an undefined term)           In geometry, a point has no dimension (actual size). Even though we represent a point with a dot, the point has no length, width, or thickness. A point is usually named with a capital letter. In the coordinate plane, a point is named by an ordered pair, (x,y). It has no dimensions; a point has no "size" it only designates location...          Collinear Points are points that lie on the same line.          Coplanar Point are points that lie on the same plane.

LINE (an undefined term)

          In geometry, a line has no thickness but its length extends in one dimension and goes on forever in both directions. A line is depicted to be a straight line with two arrowheads indicating that the line extends without end in two directions. A line is named by a single lowercase letter, , or by two points on the line, .           A line has one dimesion; a line has length (infinite) but no width or "thickness".           Opposite rays are two rays that lie on the same line, with common endpoint and no other points in common. Opposite rays form a straight line and/or a straight angle (1800).           Parallel lines are two coplanar lines that do not intersect.           Skew lines are two non-coplanar lines that do not intersect.

PLANE (an undefined term)           In geometry, a plane has no thickness but extends indefinitely in all directions. Planes are usually represented by a shape that looks like a tabletop or wall. Even though the diagram of a plane has edges, you must remember that the plane has no boundaries. A plane is named by a single letter (plane m) or by three non-collinear points (plane ABC).

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KINDS OF ANGLES

ACUTE ANGLE

An acute angle is an angle measuring between 0 and 90 degrees.

RIGHT ANGLE

     A right angle is an angle measuring 90 degrees. Two lines or line segments that meet at a right angle are said to be perpendicular. Note that any two right angles are supplementary angles (a right angle is its own angle supplement).

OBTUSE ANGLE

     An obtuse angle is an angle measuring between 90 and 180 degrees.

Example:

COMPLEMENTARY ANGLES

     Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. One of the complementary angles is said to be the complement of the other.

     Notice that these two angles can be "pasted" together to form a right angle!

SUPPLEMENTARY ANGLES

Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees. One of the supplementary angles is said to be the supplement of the other.

Notice that these two angles can be "pasted" together to form a straight line!

For any two lines that meet, such as in the diagram below, angle AEB and angle DEC are called vertical angles. Vertical angles have the same degree measurement. Angle BEC and angle AED are also vertical angles.

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Answer:

Plane

plane is a flat surface that contains infinitely many intersecting lines that extend forever in all directions. Think of a plane as a huge sheet of paper with no thickness that goes on forever. We can use point, line, and plane to define new terms.

Video Definition Point Line Plane Set Examples

A certain famous, fictional spy always describes his favorite beverage as "shaken and not stirred." Four concepts in geometry can best be thought of as "described and not defined."

Undefined Terms Definition

In all branches of mathematics, some fundamental pieces cannot be defined, because they are used to define other, more complex pieces. In geometry, three undefined terms are the underpinnings of Euclidean geometry:

A fourth undefined term, set, is used in both geometry and set theory.

Even though these four terms are undefined, they can still be described. Mathematicians use descriptions of these four terms and work up from them, creating entire worlds of ideas like angles, polygons, Platonic solids, Cartesian graphs, and more.

Simply because these terms are formally undefined does not mean they are any less useful or valid than other terms that emerge from them. These four undefined terms are used extensively in theorems, proofs, and defining other words.

Point

A point in geometry is described (but not defined) as a dimensionless location in space. A point has no width, depth, length, thickness -- no dimension at all. It is named with a capital letter: Point A; Point B; and so on.

Points in geometry are more like signal buoys on the vast, infinite ocean of geometric space than they are actual things. They tell you where a spot is, but are not the spot itself (even though we show them with a dot).

When you place two points on a plane, you can create a line.

Line

A line is described (not defined) as the set of all collinear points between and extending beyond two given points. A line goes out infinitely past both points, but in geometry we symbolize this by drawing a short line segment, putting arrowheads on either end, and labeling two points on it. The line is then identified by those two points. It can also be identified with a lowercase letter.

With only three points, you can create three different lines, and you can also describe a plane.

Plane

A plane is described as a flat surface with infinite length and width, but no thickness. It cannot be defined. A plane is formed by three points. For every three points in space, a unique plane exists.

A symbol of a plane in geometry is usually a trapezoid, to appear three-dimensional and understood to be infinitely wide and long. A single capital letter, or it can be named by three points drawn on it.

Modeling a plane in everyday life is tricky. Nothing will accurately substitute for a plane, because even the thinnest piece of paper, cookie sheet, or playing card still has some thickness. Also, all of these objects end abruptly at their edges. Planes do not end, and they have no thickness.

Set

A set can be described as a collection of objects, in no particular order, that you are studying or mathematically manipulating. Sets can be all these things:

• Physical objects like angles, rays, triangles, or circles
• Numbers, like all positive even integers; proper fractions; or decimals smaller than 0.001
• Other sets, like the set of all even numbers and the set of all multiples of five; the set of all acute angles and the set of all angles less than 15°

In geometry, we use sets to group numbers or items together to form a single unit, like all the triangles on a plane or all the straight angles on a coordinate grid. Sets are shown by using braces, { } on either side of the set:

• { 0.1, 0.2, 0.3 } for a set of three decimal numbers
• { 1, 2, 4, 8, 16 … } for the infinite set of powers of two
• {acute angles, obtuse angles, reflex angles, straight angles} for a set of angles found in plane geometry
• { 1, 2, 3 … } for the infinite set of whole positive integers
• { A, B, C … X, Y, Z } for the set of English alphabet letters

A set does not need to have a limit. The ellipsis ( … ) can indicate more terms between the start and end of the series, or it can indicate that the set between the braces continues on, infinitely.

A set does not need to ordered, like an array. You can write the first two sets shown above like this:

• { 0.2, 0.1, 0.3 } or like this { 0.3, 0.2, 0.1 }
• { 4, 8, 16, 1, 2 … } or like this { 16, 2, 4, 1, 8 ... }

Undefined Terms Examples

Look on the floor of your bedroom. Mentally arrange a set of what you see. It might look like this:

• { socks, gym shorts, left shoe, geometry textbook }

Look at a calendar. Mentally (or, better, jot down) a set of Saturday and Sunday dates. It might look like this:

• { 13, 14, 6, 20, 7, 27, 21, 28 }

The order does not matter, but the set might be easier to work with in order from least to greatest:

• { 6, 7, 13, 14, 20, 21, 27, 28 }

Lesson Summary

Now that you have navigated your way through this lesson, you are able to identify and describe three undefined terms (point, line, and plane) that form the foundation of Euclidean geometry. You can also identify and describe the undefined term, set, used in geometry and set theory.

Next Lesson:

Geometric Figures