Question

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Answer 1

You Answer In Attachment

Hope it's Helpful

Answer 2

Given

• POQ is a line.
• OR is perpendicular to line PQ.
• OS lies between OP and OR.
• ∠ROQ = 90°

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To Prove

• ROS = ¹/₂ (∠QOS - ∠POS)

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Proof

Since POQ is a line all angles would sum upto 180°

⇒ ∠POS + ∠ROS + ∠ROQ = 180° [Linear Pair]

⇒ ∠POS + ∠ROS + 90° = 180°

⇒ ∠POS + ∠ROS = 180° - 90°

⇒ ∠POS + ∠ROS = 90°

⇒ ∠ROS = 90° - ∠POS      ____ (i)

From the figure,

⇒ ∠ROS + ∠ROQ = ∠QOS

⇒ ∠ROS + 90° = ∠QOS

⇒ ∠ROS = ∠QOS - 90°      ____ (ii)

Adding (i) and (ii)

⇒ ∠ROS + ∠ROS = 90° - ∠POS + ∠QOS - 90°

⇒ 2 ∠ROS = 90° - 90° + ∠QOS - ∠POS

⇒ 2 ∠ROS = ∠QOS - ∠POS

⇒ ∠ROS = $\bf \dfrac{1}{2}( \angle QOS - \angle POS)$

Hence, proved.

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