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Three angles ∠P, ∠Q, and ∠R of a ∆PQR are 60°, 80°, and 40° respectively. The external angle bisector of ∠R meet at S with internal angle bisector of ∠Q and I is the incentre of ∆QRS, then what is the measure of ∠QIR?

Three angles ∠P, ∠Q, and ∠R of a ∆PQR are 60°, 80°, and 40° respectively. The external angle bisector of ∠R meet at S with internal angle bisector of ∠Q and I is the incentre of ∆QRS, then what is the measure of ∠QIR? Correct Answer 105°

Given:

∠P, ∠Q, and ∠R of a ∆PQR are 60°, 80°, and 40° respectively

Calculation:

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∠P = 60°, ∠Q = 80° and ∠R = 40°.

∠PRS = (180° – 40°)/2 = 70°

∠SQR = 80°/2 = 40°

In ∆SQR-

∠QRS = 40° + 70° = 110°

⇒ ∠QSR = 180° – 40° – 110° = 30°

∠QIR = 90° + (∠QSR/2)

⇒ ∠QIR = 90° + 15°

∴ ∠QIR = 105°

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