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In the figure given below, let POQ is a line and ray OR is perpendicular to line PQ.

Let OS is another ray lying between rays OP and OR. What is the value of $∠ ROS$ ?

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12 (∠ QOS − ∠ POS)

12 (∠ P OS − ∠ Q OS)

12 (∠ QOS+ ∠ POS)

None of these

answer is A.

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Detailed Solution

Given Data,

O R ⊥ P Q

and OS is another ray lying between the rays OP and OR.

∴ ∠ R O Q = 90

In fig. POQ is a line . Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR . prove that ROS = 1/2( QOS - POS).

Since POQ is a straight line, so the sum of all angles made on it is 180°.

∴ ∠ P O S + ∠ R O S + ∠ R O Q = 180 ° ... 1

⇒ ∠ P O S + ∠ R O S + 90 ° = 180 ° ⇒ ∠ P O S + ∠ R O S = 90 ° ⇒ ∠ R O S = 90 ° − ∠ P O S ... 3

From figure,

⇒ ∠ Q O S = ∠ R O S + ∠ R O Q

⇒ ∠ R O S = ∠ Q O S − ∠ R O Q .... (4)

Adding equations (3) and (4), we get,

∠ R O S + ∠ R O S = 90 ° − ∠ P O S + ∠ Q O S − 90 ° ∵ ∠ R O Q = 90 °

⇒ 2 ∠ R O S = ∠ Q O S − ∠ P O S ⇒ ∠ R O S = 12 ∠ Q O S − ∠ P O S

Hence,

∠ R O S = 12 ∠ Q O S − ∠ P O S .

So, option (1) is correct.

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