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Offline Centres

In figure, if $O P ‖ R S, ∠ O P Q = 110 °$ and $∠ Q R S = 130 °,$ then $∠ P Q R$ is equal to

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400

500

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answer is C.

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Detailed Solution

Given,

∠ O P Q = 1100

∠ Q R S = 1300

We extend the line OP to H, which intersect the line QR at A. Question Image

A transverse line across a set of parallel lines makes equal pair of corresponding angles. The straight line makes an angle

180 °

at any point in straight line.
Since RS

∥ O H

∴ ∠ Q R S = ∠ Q A

(Corresponding angles)

∠ Q A H = 1300 ..... (1)

According to the straight-line angles sum property, the sum of the angles at the point P are equal to

180 °

∠ O P Q + ∠ Q P A = 1800 1100 + ∠ Q P A = 1800 (∵ ∠ O P Q = 1100) ∠ Q P A = 1800 − 1100 ∠ Q P A = 700 … … (2)

According to the straight-line angles sum property, the sum of the angles at the point A are equal to

180 °

∠ P A Q + ∠ Q A H = 1800 ∠ P A Q + 1300 = 1800 (∵ ∠ Q A H = 1300 according to (1)) ∠ P A Q = 1800 − 1300 ∠ P A Q = 500 … .. (3)

According to the figure,

∠ P Q R

is same as

∠ P Q A

.
Hence,

∠ P Q A = ∠ P Q R

Now in

Δ P Q A

,
According to the angle sum property of a triangle, we have

∠ P Q A + ∠ Q P A + ∠ P A Q = 1800 ∠ P Q A + 700 + 500 = 1800 (from (2) and (3)) ∠ P Q A = 1800 − 1200 ∠ P Q A = 600

Hence,

∠ P Q R = ∠ P Q A = 600

Thus, measure of angle

∠ P Q R = 600

.
Therefore, option (3) is correct.

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