Plane Geometry - Bruce E. Shapiro

SECTION 16. ANGLES 85Figure 16.7:16.10).Illustration of the betweenness theorem for rays (theoremSince ray −→ AE is in the interior of angle ∠BAD, that means −→ AC must alsobe in the interior of ∠BAD.Hence C is in the interior of angle ∠BAD.Theorem 16.10 (Betweenness Theorem for Rays) Let A, B, C, andD be four distinct points such that C and D lie on the same side of ←→ AB.Thenµ(∠BAD) < µ(∠BAC) ⇐⇒ −→ −→ −→AB ∗ AD ∗ ACProof. See figure 16.7.To prove (⇐), assume −→ AB ∗−→ AD ∗−→ AC.D is in the interior of ∠BAC by Corollary 16.7.By the angle addition postulateµ(∠BAC) = µ(∠BAD) + µ(∠DAC)and by the protractor postulate ∠DAC > 0 because AB ≠ AC. Henceµ(∠BAD) < µ(∠BAC)To prove the (⇒) we will prove the contrapositive, which is∼ ( −→ AB ∗−→ AD ∗−→ AC) ⇒ [µ(∠BAC) ≤ µ(∠BAD)]So we start by assuming ∼ ( −→ −→ −→AB ∗ AD ∗ AC).Since D and C are on the same side of ←→ AB we must rule out the possibilitythat −→ −→ −→AD ∗ AB ∗ ACRevised: 18 Nov 2012 « CC BY-NC-ND 3.0.