Question

Using the hyperbolic angle sum theorem, prove theorem 1. theorem 1 (the exterior angle inequality theorem): An exterior angle of a triangle is greater than each of the remote/nonadjacent interior angles of the triangle. hyperbolic angle sum theorem: The sum of the measures of the angles of a hyperbolic triangle is less than 180°. Please give an original answer, the explanations already posted on Chegg do not answer this question or do not answer in a language that I understand (such as, h i inversion, i mapping, etc.) Thanks!

Answer #1

Use the Exterior Angle Theorem to show that the sum of the
measures of two interior angles of a triangle is always less than
180 degrees.
Can not use that the sum of angles is 180 degrees.

Assuming Playfair’s axiom. Prove the Alternate interior angle
theorem and that the sum of the interior angles if a triangle is
180. Use these results to solve problems. Please show your work and
explanations.

Describe a spherical triangle in spherical geometry that is a
counterexample to theorem 1.
Explain how the spherical triangle proves theorem 1 is false in
spherical geometry.
theorem 1
theorem 1 (the exterior angle inequality
theorem): An exterior angle of a triangle is greater than
each of the remote/nonadjacent interior angles of the triangle.

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