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Searching for ## Fundamental Axioms and Theorems

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Geometry

You will get to learn the fundamental ideas you need to understand in order to make sound geometric conjectures and prove them.

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After completing this tutorial, you will be able to complete the following:

- Know that a line is a collection of points.
- Know that planes can be viewed either as a collection of lines or as a collection of points.
- Define intersecting, parallel, and coincident lines.

Consider the following diagram:

By considering the line d and the plane P as sets, how can we describe the relationship between point A and the plane P and between line d and the plane P?

~ The point A is one of the points in the plane P, so we can say that A is an element of the set of points that make up the plane P. We say that A is in P. Line d is made up of points that are in the plane P, so line d is a subset of the set of points of P. We also say that d is in P.

What sorts of relationships are there between points and lines on the same plane?

~ Given a point and a line in the same plane, the point is either on the line or not on the line. In the diagram below, point A is not on line d, but point B is on line d. Now B is one of the points that make up the line d, so B is an element of the set of points that are on line d. In this case, we say that B is in d.

What sorts of relationships are there between two lines on the same plane?

~ There are only three possibilities. Two lines on the same plane may intersect at exactly one point, two lines on the same plane may be coincident (if they intersect at infinitely many points), or two lines on the same plane may be parallel (if they do not intersect). In the diagram below, lines l and d intersect at exactly one point, lines n and d are coincident, and lines m and d are parallel.

Approximate Time | 2 Minutes |

Pre-requisite Concepts | Students should be familiar with coincident lines, element of a line, and intersecting lines. |

Course | Geometry |

Type of Tutorial | Animation |

Key Vocabulary | coincident lines, element of a line, intersecting lines |