Question : In triangle PQR, the sides PQ and PR are produced to A and B respectively. The bisectors of $\angle {AQR}$ and $\angle {BRQ}$ intersect at point O. If $\angle {QOR} = 50^{\circ}$ what is the value of $\angle {QPR}$ ?
Option 1: $50^{\circ}$
Option 2: $60^{\circ}$
Option 3: $80^{\circ}$
Option 4: $100^{\circ}$
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Correct Answer: $80^{\circ}$
Solution :
In triangle PQR, the bisectors of $\angle {AQR}$ and $\angle {BRQ}$ intersect at point O. According to the Angle bisector theorem, the angles formed at the incenter by the angle bisectors are half the sum of the other two angles of the triangle.
$\angle {QOR}= \frac{180^{\circ} - \angle {QPR}}{2}$
⇒ $50^{\circ}= \frac{180^{\circ} - \angle{QPR}}{2}$
⇒ $\angle {QPR} = 180^{\circ} - 2 \times 50^{\circ} = 80^{\circ}$
Hence, the correct answer is $ 80^{\circ}$.
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