In the following figure, $∠ P Q R = 60^{\circ}, ∠ P Q R$ is bisected and the resultant angles are bisected again.

Find $∠ T Q S + ∠ S Q U + ∠ P Q S .$

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Solution

The correct option is C

$G i v e n : ∠ P Q R = 60^{\circ} ∠ R Q T = ∠ T Q P (QT bisects angle RQP) \Rightarrow ∠ R Q T = ∠ P Q R / 2 = \frac{60^{\circ}}{2} = 30^{\circ} Similarly, ∠ R Q U = ∠ U Q T = \frac{30^{\circ}}{2} = 15^{\circ} (Q U b i s e c t s ∠ R Q T) A n d, ∠ T Q S = ∠ Q S P = \frac{30^{\circ}}{2} = 15^{\circ} (Q S b i s e c t s ∠ T Q P) S o, ∠ T Q S + ∠ S Q U + ∠ P Q S = i .) ∠ T Q S = 15^{\circ} i i .) ∠ S Q U = ∠ U Q T + ∠ T Q S = 15^{\circ} + 15^{\circ} = 30^{\circ} i i i .) ∠ P Q S = 15^{\circ} \Rightarrow ∠ T Q S + ∠ S Q U + ∠ P Q S = 15^{\circ} + 30^{\circ} + 15^{\circ} = 60^{\circ}$

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