Theorem 1: An exterior angle of a triangle is greater than either of the nonadja

Question

Theorem 1: An exterior angle of a triangle is greater than either of the nonadjacent interior angles of the triangle.

A. Using the Euclidean angle sum theorem, prove Theorem 1. Your proof must refer to the definitions provided below.

Definitions:

• adjacent: Two angles are adjacent if they share a common vertex and common side, and they do not overlap. Otherwise, the two angles are nonadjacent.

• supplementary:Two angles are supplementary if their measures sum to 180°.

• exterior:An angle that is both adjacent and supplementary to an angle of a triangle is an exterior angle of the triangle.

1. Explain why this theorem is also true in hyperbolic geometry.

Here is what i have in referance to A2 and i need help understanding a1:

A2: The triangle exterior angle theorem is invalid in spherical geometry or is not in the associated elliptical geometry. We will use a rounded triangle with one of its vertices being the North Pole and the extra two existing about the equator. The legs of the triangle originate from the North Pole (great rings of the Earth) with the pair joining the equator at right angles, so the triangle has an external angle that is equivalent to an obscure internal angle. The other internal angle (at the North Pole) can be made bigger than 90°, more affirming the inaccuracy of this remark. Nevertheless, seeing that the Euclid's triangle exterior angle theorem is a theorem in absolute geometry it is naturally conclusive in hyperbolic geometry. The proof is already written i just need to know why this is also true in hyperbolic geometry, section 1.

Explanation / Answer

A. According to Euclid's sum, A + B + C = 180. Where A and B are the two non-adjacent angles in question. Let D be the exterior angle. Then C + D =180.

=> A + B = D.

Hence D>A and D>B.

A1. In hyperbolic geometry, A + B + C <180.

=> A + B < 180 -C

=> A + B < D

=> D>A and D>B

A2. In Spherical (and thereby elliptical) geometry,

A + B + C > 180

=> A + B > 180-C

=> A + B > D

Hence we cannot come to the usual conclusions.

[Please note that a hyperbolic triangle is CONVEX and has a positive defect while spherical triangle is CONCAVE which results in a negative defect. This explains the sum of the angles being less than or greater than 180]

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