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 ∠ROS = 1/2 (∠QOS - ∠POS).
Solution:
Given: OR is perpendicular to PQ. ∠ROQ = ∠ROP = 90°.
To prove: ∠ROS = 1/2(∠QOS - ∠POS).
When a ray intersects a line, the sum of adjacent angles so formed is 180°.
Let ∠ROS = a, ∠POS = b and ∠SOQ = c.
To prove that: a = 1/2(c - b).
Since ∠ROQ = ∠ROP = 90°,
We can say, ∠POS + ∠SOR = ∠POR
b + a = 90°.….(1)
Line PQ is intersected by ray OS.
Hence ∠POS + ∠SOQ = b + c = 180°
b + c = 180°….. (2)
From equation (1), we get a + b = 90°
Multiplying by 2 on both sides we get,
2(a + b) = 2 × 90°
2(a + b) = 180°…..(3)
Comparing equations (3) and (2),
2(a + b) = b + c
2a + 2b = b + c
2a = b + c - 2b
2a = c - b
a = 1/2 (c - b)
∴ ∠ROS = 1/2 (∠QOS - ∠POS)
☛ Check: NCERT Solutions Class 9 Maths Chapter 6
Video Solution:
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 ∠ROS = 1/2 (∠QOS - ∠POS).
NCERT Solutions Class 9 Maths Chapter 6 Exercise 6.1 Question 5
Summary:
In Fig. 6.17, if POQ is a line OR is perpendicular to PQ, and OS is another ray lying between rays OP and OR, then ∠ROS = 1/2 (∠QOS − ∠POS).
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