By CK-12 Foundation

CK-12’s Geometry - moment version is a transparent presentation of the necessities of geometry for the highschool pupil. subject matters comprise: Proofs, Triangles, Quadrilaterals, Similarity, Perimeter & region, quantity, and ameliorations. quantity 1 contains the 1st 6 chapters: fundamentals of Geometry, Reasoning and evidence, Parallel and Perpendicular strains, Triangles and Congruence, Relationships with Triangles, and Polygons and Quadrilaterals.

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A segment [A, B]. note that A ≤l A ≤l A by A2 (i), while if A ≤l P ≤l A then P = A by A2 (iii). The same argument holds for ≥l . Thus [A, A] = {A}. Segments have the following properties:(i) If A = B, then [A, B] ⊂ AB. (ii) A, B ∈ [A, B] for all A, B ∈ Π. (iii) [A, B] = [B, A] for all A, B ∈ Π. (iv) If C, D ∈ [A, B] then [C, D] ⊂ [A, B]. (v) If A, B,C are distinct points on a line l, then precisely one of A ∈ [B,C], B ∈ [C, A], C ∈ [A, B], holds. Proof . 2). (i) By A1 , l = AB so [A, B] is a set of points on AB.

X + y = y where x = |A, A| and y = |A, R|. It follows that x = 0. Next with A = B let l = AB and ≤l be the natural order on l for which A ≤l B. Then we have A ≤l B, A ≤l A, |A, A| = 0, so that if we also had |A, B| = 0, then by the uniqueness part of A4 (iv) with k = 0, we would have A = B and so have a contradiction. To avoid this we must have |A, B| > 0. (ii) As P ∈ [A, B], by A4 (iii) we have |A, P|+ |P, B| = |A, B|. But by A4 (i) |P, B| ≥ 0 and so |A, P| ≤ |A, B|. If P = B, then by (i) of the present theorem |P, B| > 0 and so |A, P| < |A, B|.

The properties and proofs for degree-measure are quite like those for distance, with the role of interior regions analogous to that of segments. We note that A5 (i) is like A4 (i), A5 (iii) is like A4 (iii), A5 (iv) is like the uniqueness part of A4 (iv) and A5 (v) is like the existence part of A4 (iv). 2 Derived properties of degree-measure Deﬁnition . For a wedge-angle ∠BAC, if we take a point B1 = A so that A ∈ [B, B1 ] and a point C1 = A so that A ∈ [C,C1 ], then ∠B1 AC1 is called the vertically opposite angle of ∠BAC.