POQ is represented as a line in the following diagram. Which of the following is correct if ray OS is lying between rays OP and OR and ray OR is a ray perpendicular to line PQ?

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∠ R O S = 12 (∠ Q O S − ∠ P O S)

∠ R O S = 12 (∠ Q O S + ∠ P O S)

∠ R O S = (∠ Q O S − ∠ P O S)

∠ R O S = (∠ Q O S + ∠ P O S)

answer is A.

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Detailed Solution

It is given that POQ is a line such that a ray OR is perpendicular to line PQ and OS is another ray lying between rays OP and OS.
We know that if two lines are perpendicular, the angle subtended between them is

90 ° .

Since a ray OR is perpendicular to line PQ, therefore we have,

∠ Q O R = 90 °

∠ P O R = 90 °

From the figure, we have,

∠ Q O R = ∠ Q O S − ∠ R O S = 90 °

⇒ ∠ R O S = ∠ Q O S − 90 ° (1)

and ∠ P O R = ∠ P O S + ∠ R O S = 90 °

⇒ ∠ R O S = 90 ° − ∠ P O S (2)

Adding equations (1) and (2), we get,

2 ∠ R O S = ∠ Q O S − ∠ P O S

⇒ ∠ R O S = 12 (∠ Q O S − ∠ P O S)

Therefore, if a ray OR is perpendicular to line PQ and OS is another ray lying between rays OP and OS, then

∠ R O S = 12 (∠ Q O S − ∠ P O S) .

Hence, option (1) is correct.

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