Answers:
RQT = 159
RQS = 69
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Step-by-step explanation:
Draw a segment connecting R and S.
Triangle SRQ is a right triangle due to Thale's Theorem. That theorem is a special case of the inscribed angle theorem. The 90 degree angle R is opposite the diameter SQ.
Minor arc QR = 42 degrees. Half of this is 42/2 = 21, which is the measure of inscribed angle RSQ. This inscribed angle subtends minor arc QR.
Focus on triangle SRQ. We have two known angles (R = 90 and S = 21). Let's use them to find the missing angle.
The inside angles of any triangle always add to 180 degrees.
S+R+Q = 180
21+90+Q = 180
111+Q = 180
Q = 180-111
Q = 69
Angle RQS is 69 degrees.
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We'll add angle SQT onto the previous result to get angle RQT.
angle RQT = (angle RQS) + (angle RQT)
angle RQT = (69) + (90)
angle RQT = 159 degrees