📐 In the Realm of Neutral Geometry: What Remains True?

Where Euclid's Fifth Axiom Falls Silent, What Can We Still Trust?

(a) Parallel lines exist.

True: Parallel lines — defined as lines that never intersect — still exist even when Euclid's parallel postulate is withheld.

Justification: Both Euclidean and hyperbolic models support the existence of at least one parallel to a given line through a point not on the line.

(b) If a line \( t \) is perpendicular to distinct lines \( \overleftrightarrow{AB} \) and \( \overleftrightarrow{CD} \), then \( \overleftrightarrow{AB} \parallel \overleftrightarrow{CD} \).

False: In neutral geometry, two lines can both be perpendicular to a third and yet not be parallel.

Justification: Hyperbolic geometry allows for such a case — ultraparallel lines defy Euclidean intuition and don’t intersect, but aren’t parallel in the classical sense.

(c) Given 3 distinct collinear points, exactly one is between the other two.

True: The order of collinear points is preserved, with one point lying precisely between the other two.

Justification: SMSG’s betweenness axioms remain intact in neutral geometry, guaranteeing consistent point ordering.

(d) If \( m(\angle BAC) = m(\angle BAD) = m(\angle DAC) \), then \( D \in \text{int}(\angle BAC) \).

False: Equal angle measures do not confirm that point \( D \) lies strictly within the interior of angle \( \angle BAC \).

Justification: Point \( D \) might rest on one of the rays forming the angle — on the boundary, not the interior. Neutral geometry allows this ambiguity.