Question

In Fig. 6.17, POQ is a line. Ray OR is perpendicular
to line PQ. OS is another ray lying between rays
OP and OR. Prove that
1
ROS = -( QOS- POS).
).
2.​

Answer 1

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Answer 2

Given : -

• OR is perpendicular to PQ. ∠ROQ = ∠ROP = 90°

To prove :-

• ∠ROS = 1/2(∠QOS - ∠POS)

Solution :-

: $\implies$ Given , OR, POQ

According to questions ,

: $\implies$ ∠ POR = ∠ ROQ = 90 °

To Show ∠ ROS = 1 / 2 ( ∠ QOS - ∠ POS )

proof :-

: $\implies$∠ QOS = ∠ ROS + ∠ ROQ

: $\implies$ ∠ QOS - ∠ ROQ = ∠ ROS

: $\implies$ ∠ QOS - 90 ° = ∠ ROS - (i)

Again,

: $\implies$ ∠ POR = ∠ POS + ∠ ROS

: $\implies$ ∠ POR - ∠ POS = ∠ ROS

: $\implies$ 90° - ∠ POS = ∠ ROS - ( ii )

from (i) + (ii)

: $\implies$ ∠ QOS - 90 + 90 - ∠ POS = 2 ∠ ROS

: $\implies$ ∠ QOS - ∠ POS / 2 = ∠ ROS

: $\implies$ ∠ ROS = 1 / 2 ( ∠ QOS - ∠ POS )

Hence, proved !!