Bunuel wrote:

In the diagram below, PQ is a diameter of the circle having center at O. What is the measure of ∠PTQ?

(1) ∠ROS = 40◦.

(2) ∠RPO = 55◦.

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We have \(\widehat{POS}=180^o -2\widehat{SPO}\)

Also \(\widehat{POS}=\widehat{POR}+\widehat{ROS}\)

We also have \(\widehat{POR} = 180^o -2\widehat{RPO}\)

Hence \(\widehat{POS}=\widehat{POR}+\widehat{ROS} \\

\implies 180^o -2\widehat{SPO} = 180^o -2\widehat{RPO} +\widehat{ROS} \\

\implies \widehat{ROS} = 2\widehat{RPO} - 2\widehat{SPO} \\

\implies \widehat{ROS} = 2\widehat{RPS}\)

(1) If \(\widehat{ROS} = 40^o \implies \widehat{RPS} = 20^o\)

Since \(\widehat{PST} = 90^o \implies \widehat{PTS} = 90^o - \widehat{RPS} = 70^o\). Sufficient.

(2) We can't know the value of \(\widehat{PQT}\) so we can't know the value of \(\widehat{PTS}\). Insufficient.

The answer is A.

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