We are given a circle with diameters $WU$ and $SV$. The measure of angle $WXS$ is $75^\circ$ and the measure of angle $UXV$ is $85^\circ$. We need to find the measures of arcs $ST$, $SU$, $TUW$, and $VT$ in degrees.

1. Problem Description

We are given a circle with diameters $WU$ and $SV$. The measure of angle $WXS$ is $75^\circ$ and the measure of angle $UXV$ is $85^\circ$. We need to find the measures of arcs $ST$, $SU$, $TUW$, and $VT$ in degrees.

2. Solution Steps

Since $WU$ and $SV$ are diameters, we know that $WSU$ and $SUV$ are semicircles, and thus measure $180^\circ$.

Similarly, $WTV$ and $WUV$ are semicircles and thus measure $180^\circ$.

Arc $WS$ has the same degree measure as central angle $WXS$, so $WS = 75^\circ$.

Since $WU$ is a diameter, $WU = 180^\circ$. Then $SU = 180^\circ - WS = 180^\circ - 75^\circ = 105^\circ$.

Arc $UV$ has the same degree measure as central angle $UXV$, so $UV = 85^\circ$.

Since $SV$ is a diameter, $SV = 180^\circ$. Then $ST = 180^\circ - UV - WS = 180^\circ - (75^\circ + 85^\circ) = 180^\circ - 160^\circ = 20^\circ$.

$TU = SU - ST = 105^\circ - 20^\circ$. The measure of arc $TU$ is $85 - ST$.

Since $SU = ST + TU$, $105^\circ - UT$. $SU = 180^\circ - WS$, so $SU = 180 - 75 = 105$.

Thus, $TU = SU - ST$, and angle $SXT$ = angle $VXU$.

Since the arc ST corresponds to the angle SXT, ST is equal to UXV angle minus SXW angle.

Since $SU = ST + TU$, we have $105 = 85 + ST$. So, measure arc ST = 180 - 75 -85 =

2

0. Arc $ST = 20^\circ$.

Arc $SU = 105^\circ$.

Arc $TUW = TU + UW$. Since $TU = 85^\circ$, and $UW = WS+ST = 75+20$, so $TUW = 85 + 75 =160$, minus St which is

2

0. Arc ST is given by the $UXV - WXS = 85 - 75 = 10$. This would be not correct.

The calculation of arc $ST$ is thus:

Since $WV = 180^{\circ}$, $WS+ST+TV= 180$, $75+ ST +85 =180$, $ST = 180- (85+75)=180-160= 20$.

Arc $TU = 180 - (angle TSV)$. Since $Angle TSV = Angle USV - Angle UST = 90^{\circ} - x$

$m(TUW) = m(TU) + m(UW) = 85+75 = 160^\circ -50^\circ$

Arc $TUW = TU + UW = TU+ WS + ST$. So, $TU= SU - ST$, $TU= 105 -20 = 85$, $TUW = 85 + 75=185 -25 + 165 + 5$, $85 + 75 = 160$.

$VT = 180 - 75 - 20 -450$

Arc $VT$ has the same degree measure as central angle $VXT$, so $VT = 75$. VT = WS = 75

* $ST = 20^\circ$

* $SU = 105^\circ$

* $TUW = TU+UW$, TU = 85 and UW =

7

5. So, TUW = 160 degree.

* $VT = 75^\circ$.

3. Final Answer

ST = 20

SU = 105

TUW = 160

VT = 75