Question

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

Answer 1

Let AOB denote a straight line and let $\angle AOC\text{and}\angle BOC$ be the supplementary angles.
Then, we have:
$\angle AOC=x°\text{and}\angle BOC=\left(180-x\right)°$
Let $OE\text{bisect}\angle AOC\mathrm{and}OF\text{bisect}\angle BOC$.
Then, we have:
$$\begin{gathered} \angle AOE=\angle COE=\frac{1}{2}x°\text{and} \\ \angle BOF=\angle FOC=\frac{1}{2}\left(180-x\right)° \end{gathered}$$
Therefore,
$\angle COE+\angle FOC=\frac{1}{2}x+\frac{1}{2}\left(180°-x\right)$
$$\begin{gathered} =\frac{1}{2}\left(x+180°-x\right) \\ =\frac{1}{2}\left(180°\right) \\ =90° \end{gathered}$$