By Michal A.D., Botsford J.L.

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AXIOMS OF INCIDENCE AND ORDER. The above contradiction proves that the point R cannot lie between P and Q. Similarly, one can prove that Q does not lie between R and P , while P does not lie between Q and R. Thus, none of the points P , Q, and R on the line a lies between two others. 4. Therefore the initial assumption that the line a crosses both segments [AC] and [BC] at their interior points is invalid. 5 is proved. § 3. Segments on a straight line. 1. Let A, B, C, and D be a group of four points.

1998, 2007. 64 CHAPTER III. AXIOMS OF CONGRUENCE. (3) for any element a ∈ G there is an element a′ such that a · a′ = a′ · a = e. The element e is called the unity of the group G, while the element a′ is called the inverse element for an element a ∈ G. 3. A group G is called commutative or Abelian if the group multiplication in it is commutative, i. e. a · b = b · a. 4 shows that the set of slipping vectors is an Abelian group with respect to the addition. The multiplication sign in it is replaced by the plus sign, while the zero vector plays the role of the unity.

For this reason we shall denote such an angle as ∠hOk or even as ∠hk. Axiom A16. Any angle ∠hk is congruent to itself and for any half-plane a+ with a ray m lying on the boundary line a there is a unique ray n within the half-plane a+ such that ∠hk ∼ = ∠mn. Axiom A17. Let A, B, and C be three points not lying on one ˜ B, ˜ and C˜ be other three points also not straight line and let A, lying on one straight line. If the conditions ˜ [AB] ∼ = [A˜B], ˜ [AC] ∼ = [A˜C], ˜ A˜C˜ ∠BAC ∼ = ∠B ˜ C˜ and are fulfilled, then the other two conditions ∠ABC ∼ = ∠A˜B ˜ are also fulfilled.