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Euclidean geometry axioms

Axiom of belonging. Axiom of ordering.
Axiom of congruence (equality) of segments and angles.
Axiom of parallel straight lines.
Archimedean axiom of continuity.

As we have noted above, there is a set of the axioms – properties, that are considered in geometry as main ones and are adopted without a proof . Now, after introducing some initial notions and definitions we can consider the following sufficient set of the axioms, usually used in plane geometry.

Axiom of belonging. Through any two points in a plane it is possible to draw a straight line, and besides only one.

Axiom of ordering.  Among any three points placed in a straight line, there is no more than one point placed between the two others.

Axiom of congruence  ( equality )  of segments and angles. If two segments (angles) are congruent to the third one, then they are congruent to each other.

Axiom of parallel straight lines. Through any point placed outside of a straight line it is possible to draw another straight line, parallel to the given line, and besides only one.

Axiom of continuity  ( Archimedean axiom ). Let AB and CD be two some segments;  then there is a finite set of such points  A1  ,  A2  , … , An  ,  placed in the straight line AB, that segments AA1  , A1A2  , … , An - 1An  are congruent to segment CD,  and point B is placed between A and An .

We emphasize, that replacing one of these axioms by another, turns this axiom into a theorem, requiring a proof. So, instead of the axiom of parallel straight lines we can use as an axiom the property of triangle angles (“the sum of triangle angles is equal to 180 deg”). But then we should to prove the property of parallel lines.


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