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In the figure given below, if $P R > P Q$ and PS bisects $∠ Q P R$ , then which of the following is correct?

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∠ P S R > ∠ P S Q

∠ P S R < ∠ P S Q

∠ P S R = ∠ P S Q

None of these

answer is A.

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Detailed Solution

Given that in the following figure,

P R > P Q

and PS bisects

∠ Q P R .

⇒ ∠ Q P S = ∠ R P S (1)

We know that the angle opposite to the larger side is large.

∵ P R > P Q

∴ ∠ P Q R > ∠ P R Q

⇒ ∠ P Q S > ∠ P R S

⇒ ∠ P Q S − ∠ P R S > 0 (2)

The sum of all angles of a triangle is

180 ° .

From

Δ P S Q

, we get,

∠ P S Q + ∠ S Q P + ∠ Q P S = 180 °

From

Δ P S R

, we get

∠ P S R + ∠ S R P + ∠ R P S = 180 °

Equating equation (3) and (4), we get

∠ P S Q + ∠ S Q P + ∠ Q P S = ∠ P S R + ∠ S R P + ∠ R P S

⇒ ∠ P S Q + ∠ P Q S = ∠ P S R + ∠ P R S [using eq . (1)]

⇒ ∠ P S R − ∠ P S Q = ∠ P Q S − ∠ P R S > 0 [using eq . (2)]

⇒ ∠ P S R − ∠ P S Q > 0

∴ ∠ P S R > ∠ P S Q

Hence, if

P R > P Q

and PS bisects

∠ Q P R

, then

∠ P S R > ∠ P S Q

.
Thus, option (1) is correct.

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