Primitive Notions and Propositions
[justify][b]Axioms [/b](or postulates) are unquestionably universally valid [b]truths[/b], often used as principles in the construction of a theory or as the basis for an argument. That is, an axiom is a proposion which is so clear, that is assumed as true without a demonstration or proof. An axiomatic system is the set of axioms that define a given theory and that constitute the simplest truths from which the new results of that theory are demonstrated. What this means is that for every theorem in math, there exists an axiomatic system that contains all the axioms needed to prove that theorem. The axiom contains evidence in itself and therefore does not need not be proved.[/justify]
Axiom 1- Whatever the line, there are points that [b]belong to [/b]this line, and points [b]not belonging[/b] to it.
Axiom 2- Given two distinct points, there is a [b]single [/b]line that contains them. [br][b]Note:[/b] when two lines have a point in common, they are said to intersect or cut at that point.
In the previous Geogebra applet, select the Line option (window 3) and draw a line that passes through points B and D. Draw another line that passes through D and C. Is this last line in the same plane of the other two lines?
Two distinct lines either do not intersect or intersect at [b]a single [/b]point.
Which pairs of lines intersect? Do lines g and h intersect?
[justify][b]Hypothesis: [/b]Consider [i]m[/i] and [i]n [/i]as two distinct lines.[br][b]Thesis: [/b][i]m [/i]and [i]n [/i]do not intersect or intersect at a single point. The intersection of these two lines cannot contain two or more points, otherwise, when looking at the truth provided by axiom 2 ([i]Given two distinct points there is a [b]single [/b]line that contains them[/i]), they would coincide. Therefore the intersection of [i]m [/i]and [i]n [/i]either happens at only one point or it doesn't happen. [/justify]
Did you understand the proof? Can you explain it in another way? If you didn't understand, write what you didn't understand.
Given three distinct points on a line, [b]one and only one[/b] of them is located between the other two.
The set consisting of two points [b]A[/b] and[b] B[/b] and all points between [b]A[/b] and [b]B[/b] is called line segment [b]AB[/b].[br]Points [b]A[/b] and [b]B[/b] are called line segment endpoints.
In the previous construction, select the Segment tool (window 3) and create the line segments [b]BC[/b], [b]CE [/b]and [b]AC[/b]. If you didn't understand, write what you didn't understand.
Consider [b]A [/b]and [b]B [/b]as two distinct points. The set consisting of points of segment [b]AB [/b]and of [b]all points C [/b] so that [b] B [/b]is between [b]A [/b]and [b]C, [/b]is called a ray [b]starting [/b]at [b]A [/b]passing through[b] B[/b].
In reference books, in general, the ray is represented only with an "Vector".
Note that two points [b]A [/b]and [b]B [/b]determine two rays:[sub] [/sub] S[sub]AB [/sub]and S[sub]BA[/sub].
Consider two points [b]A[/b] and [b]B[/b] of a line. There is always a point [b]C[/b] between [b]A[/b] and [b]B[/b] , and a point [b]D[/b] so that [b]B[/b] is between [b]A[/b] and [b]D.[/b]
The result in this axiom is that, between any two points on a line, there is an infinity of points.
Consider [b][i]m[/i][/b] as a line and [b]A[/b] a point that does not belong to line [b][i]m[/i][/b]. The set consisting of [b][i]m[/i][/b] and all points [b]B[/b], such that [b]A[/b] and [b]B [/b]are on the same side of line [b][i]m[/i][/b], is called a half-plane determined by [b][i]m [/i][/b]containing [b]A.[/b]
A line [i]m[/i] outlines two distinct half-planes whose intersection is line [i]m.[/i]