Question

Given: or is perpendicular to pq or and os are rays to pq to prove: ∠ros=1/2(∠qos−∠pos) proof: ∠roq+∠rop=180° (linear pair) ⇒∠rop=180°−∠roq=180°−90°=90° r.H.S =1/2(∠qos−∠pos) =12(180°−∠pos−∠pos) (∠pos+∠qos=180°) (linear pair) =12(180°−2∠pos) ...........Eq (1) we have ∠pos=∠rop−∠ros=90°−∠ros, putting this in eq(1), we get r.H.S =12(180°−2(90°−∠ros)) =1/2(180°−180°+2∠ros)=1/2(2∠ros) =∠ros therefore, l.H.S=r.H.S hence proved.

Answer 1

Answer:

 In the adjoining figure, POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that  

Ans.  

Given,

OR is perpendicular to line PQ

To prove,

∠ROS = 1/2(∠QOS – ∠POS)

Proof

∠POR = ∠ROQ = 90° (Perpendicular)

From the figure, it is clear that

∠QOS = ∠ROQ + ∠ROS = 90° + ∠ROQ — (a)

∠POS = ∠POR – ∠ROS = 90° – ∠ROQ — (b)

Subtracting (b) from (a)

∠QOS – ∠POS = 90° + ∠ROQ – (90° – ∠ROQ)

⇒ ∠QOS – ∠POS = 90° + ∠ROQ – 90° + ∠ROQ

⇒ ∠QOS – ∠POS = 2∠ROQ

⇒ ∠ROS = 1/2(∠QOS – ∠POS)

Hence, proved.