Sum of Angles between Straight Lines at Point form Four Right Angles

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Corollary to Two Angles on Straight Line make Two Right Angles

If any number of straight lines are drawn from a given point, the sum of the consecutive angles so formed is $4$ right angles.


Let $OA_1, OA_2, \ldots, OA_n$ be straight lines drawn from a point $O$ to points $A_1, A_2, \ldots, A_n$.

Let $OA_1$ be produced past $O$ to $B$.

Then $OB$ either coincides with $OA_j$ for some $j$ between $1$ and $n$, or $OB$ divides angle $A_j O A_k$ for some $j, k$ between $1$ and $n$.

First suppose $OB$ coincides with $OA_j$.