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Transcript
3.4 Congruence in the hyperbolic plane Theorem: If three angles of one triangle are congruent respectively to three angles of another triangle, the triangles are congruent (AAA congruence). Note: In the hyperbolic plane, you cannot have similarity without congruence. Theorem: Saccheri quadrilaterals with congruent summits and summit angles are congruent. Theorem: Two omega triangles are congruent if the sides of finite length are congruent and if a pair of corresponding angles not located at the omega points are congruent. Theorem: Two omega triangles AB and AB are congruent if A A and B B . Exercises 1. Given omega triangles AC and BC with AC CB , is AC BC . Justify. 2. If the angles with ordinary vertices in an omega triangle are congruent, then the line from the omega vertex to the midpoint of the opposite side is perpendicular to that side. 3. Is AAAA is a congruence condition for quadrilaterals in hyperbolic space? 4. Find the flaw in the proof of exercise # 5. page 130.