how do I prove that $∡ (ℓ, m) + ∡ (m, n) = ∡ (ℓ, n)$

Question

For all concurrent lines ℓ, m, and n, regardless of configuration. We define the directed angle to be the measure of the angle starting from ℓ and ending at m, measured counterclockwise. So I came across definition of directed angles and it really seems counterintuitive to me, I'm struggling to prove this following formula $$∡ (ℓ, m) + ∡ (m, n) = ∡ (ℓ, n)$$ as I said it seems incredibly easy thing to do, but not for stupid person like I am. It seems that I need to check so many various cases.

And in 3-dimensions I would never think that such a statement was true.

Answer 1

Actually there are only two cases to be considered (ignoring degenerate cases when some lines coincide):

1. Starting rotating counterclockwise from $l$ we meet $m$ first.
2. Starting rotating counterclockwise from $l$ we meet $n$ first. In this case the sum $\angle(l,m)+\angle(m,n)$ exceeds $180^{\circ}$ and when we take it modulo $180$ (I saw in the link that we should do it) we get what we want.