Answer:
The following measurements are:
[tex]m\angle{STR}=23^\circ[/tex] (Option #4)
[tex]m{QT}=142^\circ[/tex] (Option #7)
[tex]mST=134^\circ[/tex] (Option #5)
[tex]mRQ=38^\circ[/tex] (Option #2)
Step-by-step explanation:
To begin, we can find the measure of [tex]\angle{STR}[/tex] by applying the inscribed angle theorem: an angle θ inscribed in a circle is half of the central angle 2θ that subtends the same arc on the circle.
Since the intercepted arc (RS) is 46 degrees, we have:
[tex]46=2\theta\\23=\theta[/tex]
Next, we can find the measure of arc QT using the same theorem. So,
[tex]QT=2(71)\\QT=142[/tex]
Notice that the chord RT is actually a diameter. From the theorem about the inscribed angle including a diameter, we know that the intercepted arc will have a measure of [tex]180^\circ[/tex]. Since the arc ST is part of the arc RST, and we know RS is [tex]46^\circ[/tex], we can set up and solve this equation:
[tex]RST = RS + ST\\180 = 46 + ST\\134 = ST[/tex]
We can use the same idea to find RQ. We know that RQT is [tex]180^\circ[/tex] and QT is [tex]142^\circ[/tex], so:
[tex]RQT = RQ + QT\\180 = RQ + 142\\38 = RQ[/tex]
