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

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.

Proof

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 $OA_j$.

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Sources

• 1968: M.N. Aref and William Wernick: Problems & Solutions in Euclidean Geometry ... (previous) ... (next): Chapter $1$: Triangles and Polygons: Theorems and Corollaries $1.1$: Corollary $1$