Problem — Consecutive supplementary angles are adjacent

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Let $\widehat{POQ}$ and $\widehat{QOR}$ be two consecutive angles, that is, sharing the vertex $O$ and the side $OQ$, and lying on opposite sides of $OQ$. Given that they are supplementary, prove that they are adjacent, i.e. that the sides $OP$ and $OR$ are opposite rays.

Solution

By hypothesis $\widehat{POQ} + \widehat{QOR} = 180^\circ$.

Let $OP'$ be the ray opposite to $OP$. The rays $OP$ and $OP'$ form a straight angle, so the angles $\widehat{POQ}$ and $\widehat{QOP'}$ are supplementary: $\widehat{QOP'} = 180^\circ - \widehat{POQ}.$

On the other hand, from the hypothesis, $\widehat{QOR} = 180^\circ - \widehat{POQ}.$

Therefore $\widehat{QOR} \cong \widehat{QOP'}$. The rays $OR$ and $OP'$ both lie on the same side of $OQ$ (the side opposite to $OP$) and form congruent angles with $OQ$: by the uniqueness of the angle of given measure on a fixed side, they coincide, that is $OR \equiv OP'$.

Hence $OR$ is the ray opposite to $OP$: the two angles are adjacent. $\blacksquare$