Questions are followed by two statements labelled as I and II. Decide if these statements are sufficient to conclusively answer the question. Choose the appropriate answer from the options given below:
A. Statement I alone is sufficient to answer the question.
B. Statement II alone is sufficient to answer the question.
C. Statement I and Statement II together are sufficient, but neither of the two alone is sufficient to answer the question.
D. Either Statement I or Statement II alone is sufficient to answer the question.
E. Both Statement I and Statement II are insufficient to answer the question
In the trapezoid PQRS, PS is parallel to QR. PQ and SR are extended to meet at A. What is the value of $$\angle$$PAS ?
I. PQ = 3, RS = 4 and $$\angle$$ QPS = 60°.
II. PS = 10, QR = 5.
In the figure, $$\triangle AQR \sim \triangle APS$$
=> $$\frac{AQ}{AP} = \frac{QR}{PS} = \frac{AR}{AS} = k$$ --------Eqn(I)
Statement I : PQ = 3 cm , RS = 4 cm , $$\angle$$ QPS = 60°
In right $$\triangle$$ PQM
=> $$sin 60^{\circ} = \frac{QM}{QP}$$
=> $$\frac{\sqrt{3}}{2} = \frac{QM}{3}$$
=> $$QM = \frac{3 \sqrt{3}}{2} = RN$$
Similarly, $$sin (\angle RSN) = \frac{3 \sqrt{3}}{8}$$
=> $$\angle RSN = sin^{-1} (\frac{3 \sqrt{3}}{8})$$
$$\therefore$$ In $$\triangle$$ APS
=> $$\angle PAS = 180^{\circ} - \angle APS - \angle PSA$$
=> $$\angle PAS = 120^{\circ} - sin^{-1} (\frac{3 \sqrt{3}}{8})$$
Thus, statement I alone is sufficient.
Statement II : PS = 10, QR = 5
From eqn(I), $$k = \frac{1}{2}$$
But, we do not know anything regarding the measures of the remaining sides or any of the angles.
So, statement II is not sufficient.
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