Let $X$ be the set of lines in the Euclidean plane. For $r,s$ $\in$ $X$, $d (r, s)$ is the Euclidean usual distance between $r$ and $s$ if the two lines are parallel, otherwise the arc angle (with value in $[0,\pi/2]$) of the smallest angle formed by rays if the two lines are incident. Than $(X,d)$ is a metric space?
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HINT: What if $r$ and $s$ are parallel lines one unit apart, and $t$ is a line that makes a very small angle with $r$ and $s$? Do you have the triangle inequality $d(r,s)\le d(r,t)+d(t,s)$?
The triangle inequality does not hold. Take two parallel lines r and s and a line with very little slope with respect to them...