By Sharipov R.A.

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Therefore ∠AOB and ∠BOA are different notations for the same angle. 3). The intersection of the open half-planes a+ ∩ b+ is called the interior of the angle ∠AOB. The point O determines the division of the lines a and b into four closed half-lines (four rays). We denote them as follows: [OA , [OB , [OC , [OD . The previous notations [O, +∞) and [O, −∞) for rays are conve- 44 CHAPTER II. AXIOMS OF INCIDENCE AND ORDER. nient only if we consider rays lying on one fixed line. 1. Any angle ∠AOB is the union of its interior and two rays [OA and [OB .

AXIOMS OF CONGRUENCE. Axiom A14. The binary relation of congruence for segments is transitive, i. e. [AB] ∼ = [CD] and [CD] ∼ = [EF ] imply [AB] ∼ = [EF ]. The reflexivity of the congruence of segments is stated explicitly in the axiom A13, while the transitivity of this relation forms the content of the axiom A14. Let’s prove its symmetry. 1. The binary relation of congruence for segments is symmetric, i. e. [AB] ∼ = [CD] implies [CD] ∼ = [AB]. Proof. Assume that [AB] ∼ = [CD]. Let’s apply the axiom A13 to the segment [CD] and to the ray [AB beginning at the point A.

Let’s prove its symmetry. 1. The binary relation of congruence for segments is symmetric, i. e. [AB] ∼ = [CD] implies [CD] ∼ = [AB]. Proof. Assume that [AB] ∼ = [CD]. Let’s apply the axiom A13 to the segment [CD] and to the ray [AB beginning at the point A. From this axiom we get that there is a point E on the ray [AB such that [CD] ∼ = [AE]. Since [AB] ∼ = [CD] and ∼ [CD] = [AE], due to the axiom A14 we derive [AB] ∼ = [AE]. Now we apply the axiom A13 to the segment [AB] and to the ray [AB . It says that the point E on the ray [AB such that [AB] ∼ = [AE] is unique.

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Foundations of geometry for university students and high-school students by Sharipov R.A.


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