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In ∆PQR, perpendicular bisectors of each side meet at point C while perpendiculars from each vertex meet at point O. If the sum of ∠QOR and ∠QCR is 230° and ∠PQC is 30°, then what is the measure of ∠R?

In ∆PQR, perpendicular bisectors of each side meet at point C while perpendiculars from each vertex meet at point O. If the sum of ∠QOR and ∠QCR is 230° and ∠PQC is 30°, then what is the measure of ∠R? Correct Answer 60°

Given:

The sum of ∠QOR and ∠QCR is 230° and ∠PQC is 30°

Calculation:

 

∠QOR = 180° – ∠P

∠QCR = 2∠P

Now,

∠QOR + ∠QCR = 180° – ∠P + 2∠P

⇒ 230° = 180° + ∠P

⇒ ∠P = 50°

In ∆QCR-

∠CQR = ∠CRQ

∵ ∠QCR = 2∠P = 100°

∠CQR + ∠CRQ + ∠QCR = 180°

⇒ 2∠CQR = 180° – 100° = 80°

∠CQR = 40° 

∠Q = ∠PQC + ∠CQR = 30° + 40° = 70°

∵ ∠R = 180° – ∠P – ∠Q

⇒ ∠R = 180° – 50° – 70°

∴ ∠R = 60°

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