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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