Question

Use the image to match the arc/angle measure.

Answer 1

Answer:

The following measurements are:

$m\angle{STR}=23^\circ$ (Option #4)

$m{QT}=142^\circ$ (Option #7)

$mST=134^\circ$ (Option #5)

$mRQ=38^\circ$ (Option #2)

Step-by-step explanation:

To begin, we can find the measure of $\angle{STR}$ 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:

$46=2\theta\\23=\theta$

Next, we can find the measure of arc QT using the same theorem. So,

$QT=2(71)\\QT=142$

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 $180^\circ$. Since the arc ST is part of the arc RST, and we know RS is $46^\circ$, we can set up and solve this equation:

$RST = RS + ST\\180 = 46 + ST\\134 = ST$

We can use the same idea to find RQ. We know that RQT is $180^\circ$ and QT is $142^\circ$, so:

$RQT = RQ + QT\\180 = RQ + 142\\38 = RQ$