# Intuition of angle between two planes

Let theta be the angle between two planes (see the image). I am told that ∠AKB=∠θ if and only if AK is perpendicular to MN and KB is also perpendicular to MN (where MN represents line where two planes intersect).

My question is, why does θ will equal only when the angle is created with two perpendiculars (AK and KB) on MN (which intuitively makes sense), and also why does not ∠LKB also equal to ∠θ (which intuitively does not makes sense)?

Angles whose sides are not perpendicular to the dihedral edge are, in general, different. Let $AB$ and $CB$ be perpendicular at $B$ to the edge; if $\theta=\angle ABC$ then from the cosine law we have (see diagram below): $$\cos\theta={AB^2+BC^2-AC^2\over2\,AB\cdot BC}.$$ Let now $AD$ be parallel to the edge and $\phi=\angle DBC$. Notice that $AD$ is perpendicular to plane $ABC$, hence $AD\perp AC$ and $AD\perp AB$. From the cosine law we have: $$\cos\phi={DB^2+BC^2-DC^2\over2\,DB\cdot BC}.$$ But $DC^2=AC^2+AD^2$ and $DB^2=AB^2+AD^2$, hence: $$\cos\phi={AB^2+BC^2-AC^2\over2\,DB\cdot BC}={AB\over DB}\cos\theta.$$ If follows that $|\cos\phi|<|\cos\theta|$, unless $\cos\theta=0$, i.e. $\theta=90°$.