Question

In the circle shown below T is the center of the circle, and segments PN and SQ are diameters, and segment PR is tangent to the circle. The measure of angle SQN is 66 degrees
33 degrees
42 degrees
66 degrees
48 degrees

Answer 1

Angle SQN = 66 degrees (given)
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Use the inscribed angle theorem to get

Minor Arc SN = 2*(angle SQN)
Minor Arc SN = 2*(66)
Minor Arc SN = 132 degrees
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Arc PSN is a semicircle so it is 180 degrees, meaning that...

Arc PSN = (minor arc PS) + (minor arc SN)
180 degrees = (minor arc PS) + (132 degrees)
180 = (minor arc PS) + 132
180 - 132 = (minor arc PS) + 132 - 132
48 = minor arc PS
minor arc PS = 48 degrees
central angle RTP = minor arc PS = 48 degrees
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Focus on triangle PTR; isolate angle PRT

(angle TPR) + (angle PRT) + (angle RTP) = 180
(90) + (angle PRT) + (48) = 180
(90+48) + (angle PRT) = 180
138 + (angle PRT) = 180
138 + (angle PRT) - 138 = 180 - 138
angle PRT = 42 degrees
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Final Answer: 
The measure of angle PRT is 42 degrees