Question
If two straight lines intersect each other, then prove that the ray opposite the bisector of one of the angles so formed bisects the vertically-opposite angle.
✓
Answer
Let $AB$ and $CD$ be the two lines intersecting at a point $O$ and let ray $OE$ bisect $\angle\text{AOC}.$
Now, draw a ray $OF$ in the opposite direction of $OE$, such that $EOF$ is a straight line.
Let $\angle\text{COE}=1,\angle\text{AOE}=2,\angle\text{BOF}=3$ and $\angle\text{DOF}=4.$
We know that vertically-opposite angles are equal.
$\therefore\angle1=\angle4$ and $\angle2=\angle3$ But, $\angle1=\angle2$ $[$Since OE bisects $\angle\text{AOC}]$
$\therefore\angle4=\angle3$
Hence, the ray opposite the bisector of one of the angles so formed bisects the vertically-opposite angle.
Now, draw a ray $OF$ in the opposite direction of $OE$, such that $EOF$ is a straight line.
Let $\angle\text{COE}=1,\angle\text{AOE}=2,\angle\text{BOF}=3$ and $\angle\text{DOF}=4.$
We know that vertically-opposite angles are equal.
$\therefore\angle1=\angle4$ and $\angle2=\angle3$ But, $\angle1=\angle2$ $[$Since OE bisects $\angle\text{AOC}]$
$\therefore\angle4=\angle3$
Hence, the ray opposite the bisector of one of the angles so formed bisects the vertically-opposite angle.
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