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Undefined Terms Point, Line and Plane

Undefined Terms and Intuitive Concepts of Geometry

Undefined terms:
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. Our dot can be very tiny or very large and it still represents a point. A point is usually named with a capital letter. In the coordinate plane, a point is named by an ordered pair, (x,y).
Undefined Terms Point, Line and Plane 1

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. Unless otherwise stated a line is drawn as a straight line with two arrowheads indicating that the line extends without end in both directions. A line is named by a single lowercase letter, , or by any two points on the line.
Undefined Terms Point, Line and Plane 2

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 a parallelogram. 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).
Undefined Terms Point, Line and Plane 3

Intuitive Concepts:
There are a few basic concepts in geometry that need to be understood, but are seldom used as reasons in a formal proof.

Collinear Points

points that lie on the same line.

Coplanar points

points that lie in the same plane.

Opposite rays

2 rays that lie on the same line, with a common endpoint and no other points in common. Opposite rays form a straight line and/or a straight angle (180°).

Parallel lines

two coplanar lines that do not intersect

Skew lines

two non-coplanar lines that do not intersect.

The Basic Elements of Geometry

You are familiar with some terms like triangle, square, cube, cuboid, etc. These are examples of geometrical figures. To make these geometrical figures we need to know some basic elements. These basic elements are as follows:

Point(.): A point gives an idea of a location, by making a dot by a sharp pencil on a paper. It has no length, breadth, or thickness. It has just a position and only its location can be determined.
A point is denoted by a capital letter of the alphabet like A, B, C, etc.
For example: . P (This is point P.)

Line (↔): A line is a collection of points, which can be extended endlessly on both the sides. It has length only. It has neither breadth nor thickness.
A line can be denoted in two ways
(i) Denote it by a small letter of the alphabet like line l as shown in Fig.
(ii) Mark two points (say A and B) on the line as shown in Fig. and denote it by \(\overleftrightarrow{\text{AB}}\) or line AB.

Line Segment ( ¯ ): A Line segment is a part of a line that is bounded by two distinct end points. It has a definite length but no breadth and thickness. It is the shortest distance of any two points.
A line segment from A to B is represented by seg AB or \(\overline{\text{AB}}\) or \(\overline{\text{BA}}\).
This is seg AB or \(\overline{\text{AB}}\).

Ray (→): A ray is also a part of a line which has only one end point and can be extended endlessly in one direction. A ray has no breadth or thickness.
A ray is represented by \(\overrightarrow{AB}\). It shows that A is the fixed point and B is a point on the path of a ray.
Light coming from the sun or torch is an example of a ray.

Comparison between line, line segment, and ray
Table shows the comparison between line, line segment, and ray.

Line	Line Segment	Ray
1. A line has no definite length,	A line segment has definite length.	A ray has no length.
2. A line has no end points.	A line segment has two end points.	A ray has one end point.
3. A line has no thickness.	A line segment has no thickness.	A ray has no thickness.
4. A line AB is represented by \(\overleftrightarrow{\text{AB}}\)	A line segment AB is represented by \(\overline{\text{AB}}\).	A ray AB is represented by \(\overrightarrow{AB}\).

Plane: A plane is a flat smooth surface that extends indefinitely in all directions. It has length and breadth but no thickness.
The top of a table, top and bottom of a cylinder, surface of a blackboard, etc. give the idea of a plane.
plane A plane can be denoted by taking three or more points on it, which do not lie on the same line. Plane-1

Incidence properties in a plane
The relationship between a point and a line in a plane is called the incidence property. It states that
Plane-2

Infinite number of lines can be drawn passing through a given fixed point in a plane.
Lines l₁, l₂, l₃, l₄, …… all pass through a point A.
One and only one line can be drawn passing through two given points in a plane.

If A and B are the two points in a plane, then l₂becomes the unique line that passes through the points A and B.
Plane-3

COLLINEAR POINTS
Three or more points are said to be collinear, if they lie on the same line in a plane. This line is called the line of collinearity.
In the above Fig. points A, B, and C lie on the same straight line l, so they are collinear points. If a straight line in a plane contains two points but it does not contain the third point, then these three points are said to be non-collinear. In the below Fig. A, B, C are non-collinear.
collinear-points-1