Question

1. circle r is shown. segment tv and segment ts are tangent to the circle. the measure of arc sv is 110°. what is the measure, in degrees, of angle stv? a 55° b 70° c 110° d 125°

Explanation:

Step1: Recall circle - tangent property

The radius is perpendicular to the tangent at the point of tangency. So, $\angle TVR = 90^{\circ}$ and $\angle TSR=90^{\circ}$.

Step2: Find the measure of central angle $\angle SRV$

The measure of an arc is equal to the measure of its central - angle. Given that the measure of arc $\widehat{SV}$ is $110^{\circ}$, so $\angle SRV = 110^{\circ}$.

Step3: Use the sum of interior angles of a quadrilateral

The sum of the interior angles of a quadrilateral $TSRV$ is $360^{\circ}$. Let $\angle STV=x$. Then, in quadrilateral $TSRV$, we have $\angle TVR+\angle TSR+\angle SRV+\angle STV = 360^{\circ}$. Substituting the known values: $90^{\circ}+90^{\circ}+110^{\circ}+x = 360^{\circ}$.

Step4: Solve for $\angle STV$

Combining like terms, we get $290^{\circ}+x = 360^{\circ}$. Subtracting $290^{\circ}$ from both sides gives $x=360^{\circ}-290^{\circ}=70^{\circ}$.

Answer:

B. $70^{\circ}$