Question

Prove that the bisectors of two adjacent supplementary angles include a right angle.

Answer 1

Given: $\angle$ AOC and $\angle$ COB are two supplementary angles. rays OE and OD bisect $\angle$ BOC and $\angle$ AOC respectivey.

To prove : $\angle DOE={90}^o$

Proof: $\angle AOC+\angle COB={180}^o\left(given\right)\therefore \frac{1}{2}\angle AOC+\frac{1}{2}\angle COB=\frac{1}{2}\times {180}^o\therefore \frac{1}{2}\angle AOC+\frac{1}{2}\angle COB={90}^o\Rightarrow \angle DOC+\angle COE={90}^o\left[\text{Since OD bisects}\angle \text{AOC and OE bisects}\angle \text{BOC}\right]\Rightarrow \angle DOE={90}^o$