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Section 4.3 Neutral Constructions

Figure 4.3.1. Exterior angles are larger than nonadjacent interior angles.

The construction showing that exterior angles of triangles in neutral geometry must be larger than nonadjacent interior angles is shown in Figure 4.3.1. You can work through the steps yourself by clicking on each checkbox in turn:

  1. Extend line segment \(AC\text{.}\)
  2. Bisect line segment \(BC\text{.}\)
  3. Extend line segment \(AE\text{,}\) doubling its length.
  4. Since opposite angles are congruent, SAS congruence holds between the triangles at the upper left and lower right.
  5. Corresponding parts of those triangles must therefore be congruent, demonstraing that \(\angle ABE\) sits inside of \(\angle DCE\) (as \(\angle FCE\)), and must therefore be the smaller angle.
Activity 4.3.1. Food for thought.

Which postulates were used in deriving this result? In which geometries is this result valid?

Hint

This result holds in neutral geometry, but not in spherical geometry. Can you determine which part of the argument fails in the latter case?

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