## Triangular proof

Consider the theorems below:

The following is a theorem of Euclidean geometry:

Euclidean angle sum theorem: The sum of the measures of the angles of a triangle is 180°.

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.

2. Explain why this theorem is not true in spherical geometry.

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