Show that every conclusion of Theorem 5.1.10 is false in hyperbolic geometry. Theorem 5.1.10 (Properties of...

 

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Show that every conclusion of Theorem 5.1.10 is false in hyperbolic geometry. Theorem 5.1.10 (Properties of Euclidean Parallelograms). If ABCD is a parallelogram, then 1. the diagonals divide the quadrilateral into congruent triangles (i.e., AABC = ACDA and ΔΑΒD = ΔCDB), 2. the opposite sides are congruent (i.e., AB CD and BC = AD), 3. the opposite angles are congruent (i.e., LDAB LBCD and LABC LCDA), and 4. the diagonals bisect each other (i.e., AC and BD intersect in a point E that is the midpoint of each). M Show that every conclusion of Theorem 5.1.10 is false in hyperbolic geometry. Theorem 5.1.10 (Properties of Euclidean Parallelograms). If ABCD is a parallelogram, then 1. the diagonals divide the quadrilateral into congruent triangles (i.e., AABC = ACDA and ΔΑΒD = ΔCDB), 2. the opposite sides are congruent (i.e., AB CD and BC = AD), 3. the opposite angles are congruent (i.e., LDAB LBCD and LABC LCDA), and 4. the diagonals bisect each other (i.e., AC and BD intersect in a point E that is the midpoint of each). M