Plane Geometry - Bruce E. Shapiro

SECTION 20. THE CONTINUITY AXIOM 101Figure 20.2: The continuity axiom says that the function mapping thedistance x = BD to the measure of the angle α is continuous.Theorem 20.2 (The Continuity Axiom) The functiondefined byf : [0, d] ↦→ [0, µ(∠BAC)]f(x) = µ(∠BAD)is continuous, invertible, and its inverse is continuous.Proof. Let D, E be points such thatBy the definition of betweenness, this is true iffB ∗ D ∗ E ∗ C (20.1)BD < BE < BCFurthermore, equation 20.1 is possible iffby theorem 16.8.−→AB ∗ −→ −→ −→AD ∗ AE ∗ AC (20.2)By the betweenness theorem for rays (theorem 16.10), equation 20.2 holdsiffµ(∠BAD) < µ(∠BAE) < µ(∠BAC)Thus f(x) is strictly increasing.To show that the function is onto, pick any α ∈ (0, µ(∠BAC)) and constructa ray r with angle α.Since α < µ(∠BAC), by theorem 16.10 (betweenness for rays),−→AB ∗ −→ r ∗ −→ ACRevised: 18 Nov 2012 « CC BY-NC-ND 3.0.