Question

Let $\overleftrightarrow{AB}$ and $\overleftrightarrow{CD}$ be two straight lines intersecting at O. Let $\overrightarrow{OX}$ be the bisector of $\angle BOD$. Draw $\overrightarrow{OY}$ between $\overrightarrow{OD}$ and $\overrightarrow{OA}$ such that $\overrightarrow{OY}\perp \overrightarrow{OX}$. Identify the correct option.

• A. $\overrightarrow{OY}$ bisects $\angle BOA$
• B. $\overrightarrow{OY}$ bisects $\angle DOA$
• C. $\overrightarrow{OY}$ doesn't bisect $\angle DOA$
• D. None of these

Answer 1

The correct option is B $\overrightarrow{OY}$ bisects $\angle DOA$

Given : $OX$ bisects $\angle BOD$

$\therefore \angle DOX=\angle BOX$

$\angle AOD=180°-\angle BOX$

$\angle AOD=180°-\angle BOX$

$\angle XOY=90°$

$\angle DOY=90°-\angle XOD$

$\angle AOD-\angle DOY=180°-\angle BOD-90°+\angle DOX$

$\angle AOD-\angle DOY=90°-\angle BOD+\angle DOX$

$\angle AOD-\angle DOY=90°-(\angle BOD-\angle DOX)$

$\angle AOD-\angle DOY=90°-(2\angle DOX-\angle DOX)$

$\angle AOD-\angle DOY=90°-\angle DOX$

$\angle AOD-\angle DOY=\angle DOY$ ($\because \angle DOX+\angle DOY=90°$)

$\therefore AOD=2\angle DOY$

$\therefore OY$ is the bisector of $\angle DOA$.