Skew lines

Undefined Terms Point, Line and Plane

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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 Pointspoints that lie on the same line.
Coplanar pointspoints that lie in the same plane.
Opposite rays2 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 linestwo coplanar lines that do not intersect
Skew linestwo non-coplanar lines that do not intersect.

Pairs of Lines

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Pairs Of Lines

There are four pairs of lines:

  1. Intersecting lines
  2. Parallel lines
  3. Skew lines
  4. Perpendicular lines

If we draw two lines in a plane surface, they will either intersect each other at a point or never meet each other at all.

Intersecting lines

In figure, lines AB and CD intersect each other at a point (O), such lines are called intersecting lines.
Intersecting-lines

Parallel lines

In figure, lines AB and CD never meet each other at any point, i.e., they are equidistant from each other at each and every point. Such lines are called parallel lines. Parallel lines AB and CD are
represented as AB || CD. Here, symbol ‘||’ is read as ‘is parallel to’.
Parallel-lines

Skewed lines

Two lines in two different planes that are neither parallel nor intersecting are called skewed lines.
Skewed-linesIn figure, l and m lie in different planes, so these are called skewed lines.

Perpendicular lines

Two lines in a plane, which intersect each other and form an angle of 90°, i.e., right angle are called perpendicular lines.
Perpendicular-linesIn figure, two lines AB and CD intersecting each other form an angle of 90° at point O. So, AB and CD are perpendicular to each other and is written as AB ⊥ CD. Here, symbol ‘⊥’ is read as ‘is perpendicular to’.

Concurrent lines
Two or more lines that pass through the same point are called concurrent lines and the point of intersection is called the point of concurrency.
Concurrent-linesIn figure, lines l, m, n, and o are concurrent at the point A.