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In a ∆PQR, S is any point on side QR. Internal bisector of ∠PQS and ∠PSQ meet at I1, Internal bisector of ∠PSR, and ∠PRS meet at I2. If ∠QI1S = 100°, ∠SI2R = 120°, O is orthocentre of ∆PQR, then what is the measure of ∠QOR?

In a ∆PQR, S is any point on side QR. Internal bisector of ∠PQS and ∠PSQ meet at I1, Internal bisector of ∠PSR, and ∠PRS meet at I2. If ∠QI1S = 100°, ∠SI2R = 120°, O is orthocentre of ∆PQR, then what is the measure of ∠QOR? Correct Answer 100°

Given:

∠QI1S = 100°, ∠SI2R = 120°

Calculation:

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∠QI1S = 90° + (∠QPS/2)

⇒ 100° = 90° + (∠QPS/2)

⇒ ∠QPS/2 = 10°

⇒ ∠QPS = 20°

∠SI2R = 90° + (∠SPR/2)

⇒ 120° = 90° + (∠SPR/2)

⇒ ∠SPR/2 = 30°

⇒ ∠SPR = 60°

∠P = ∠QPS + ∠SPR

⇒ ∠P = 20° + 60° = 80°

∠QOR = 180° – ∠P

⇒ ∠QOR = 180° – 80°

∴ ∠QOR = 100°

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