Question
Prove that the bisectors of two adjacent supplementary angles include a right angle.
✓
Answer
Let $AOB$ denote a straight line and let $\angle\text{AOC}$ and $\angle\text{BOC}$
be the supplementary angles.
Then, we have: $\angle\text{AOE}=\angle\text{COE}=\frac{1}{2}\text{x}^\circ\text{ and}$
$\angle\text{BOF}=\angle\text{FOC}=\frac{1}{2}(180-\text{x})^\circ$
Therefore, $\angle\text{COE}+\angle\text{FOC}=\frac{1}{2}\text{x}+\frac{1}{2}(180^\circ-\text{x})$
$=\frac{1}{2}(\text{x}+180^\circ-\text{x})$
$=\frac{1}{2}(180^\circ)$
$=90^\circ$
be the supplementary angles.
Then, we have: $\angle\text{AOE}=\angle\text{COE}=\frac{1}{2}\text{x}^\circ\text{ and}$
$\angle\text{BOF}=\angle\text{FOC}=\frac{1}{2}(180-\text{x})^\circ$
Therefore, $\angle\text{COE}+\angle\text{FOC}=\frac{1}{2}\text{x}+\frac{1}{2}(180^\circ-\text{x})$
$=\frac{1}{2}(\text{x}+180^\circ-\text{x})$
$=\frac{1}{2}(180^\circ)$
$=90^\circ$
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