In "An Introduction to Mathematical Cryptography"'s section on lattice reduction algorithms, the authors describe Gaussian lattice reduction and claim:
... the angle $\theta$ between $v1$ and $v2$ satisfies $\left| cos \theta \right| \leq \frac{\lvert v1 \rvert}{2\lvert v2 \rvert}$, so in particular $ \frac{\pi}{3} \leq \theta \leq \frac{2\pi}{3}$.
Where $v1$ and $v2$ are the output of the Gaussian lattice reduction algorithm and $\lvert v1 \rvert \leq \lvert v2 \rvert $. This result seems very close to a few trigonometric identities but I don't see why this is true in every case. Can someone shine some light on why this guarantees the reduced vectors share an angle in that range?