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

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

 

Solution:

Let $A O B$ denote a straight line and let $\angle A O C$ and $\angle B O C$ be the supplementary angles.

Then, we have:

$\angle A O C=x^{\circ}$ and $\angle B O C=(180-x)^{\circ}$

Let $O E$ bisect $\angle A O C$ and $O F$ bisect $\angle B O C$.

Then, we have:

$\angle A O E=\angle C O E=\frac{1}{2} x^{\circ}$ and

$\angle B O F=\angle F O C=\frac{1}{2}(180-x)^{\circ}$

Therefore,

$\angle C O E+\angle F O C=\frac{1}{2} x+\frac{1}{2}\left(180^{\circ}-x\right)$

$=\frac{1}{2}\left(x+180^{\circ}-x\right)$

$=\frac{1}{2}\left(180^{\circ}\right)$

$=90^{\circ}$

 

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