Problem — Bisectors of adjacent angles are perpendicular

Text

Let $\widehat{POQ}$ and $\widehat{QOR}$ be two adjacent angles (so that $OP$ and $OR$ are opposite rays). Prove that their bisectors are perpendicular, i.e. that the angle they form measures $90^\circ$.

Solution

Since the two angles are adjacent, they are supplementary: $\widehat{POQ} + \widehat{QOR} = 180^\circ.$ Set $\widehat{POQ} = \alpha$ and $\widehat{QOR} = \beta$, with $\alpha + \beta = 180^\circ$.

Let $Os$ be the bisector of $\widehat{POQ}$ and $Ot$ the bisector of $\widehat{QOR}$. By definition of bisector, $\widehat{QOs} = \frac{\alpha}{2}, \qquad \widehat{QOt} = \frac{\beta}{2}.$

The rays $Os$ and $Ot$ lie on opposite sides of $OQ$, so the angle between the two bisectors is their sum: $\widehat{sOt} = \widehat{QOs} + \widehat{QOt} = \frac{\alpha}{2} + \frac{\beta}{2} = \frac{\alpha + \beta}{2} = \frac{180^\circ}{2} = 90^\circ.$

Hence the bisectors are perpendicular: $\boxed{\widehat{sOt} = 90^\circ}$. $\blacksquare$