Question

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

Explanation:

Step1: Recall tangent - radius property

Since $TV$ and $TS$ are tangent to the circle $R$, then $RV\perp TV$ and $RS\perp TS$, so $\angle RVS = 90^{\circ}$ and $\angle RST=90^{\circ}$.

Step2: Find the measure of central angle

The measure of arc $\overset{\frown}{SV}$ is $110^{\circ}$, so the measure of central angle $\angle SRV = 110^{\circ}$.

Step3: Use the sum of angles in a quadrilateral

In quadrilateral $RSTV$, the sum of interior angles is $360^{\circ}$. Let $\angle STV=x$. Then $90^{\circ}+90^{\circ}+110^{\circ}+x = 360^{\circ}$.
Simplify the left - hand side: $290^{\circ}+x = 360^{\circ}$.
Solve for $x$: $x=360^{\circ}- 290^{\circ}=70^{\circ}$.

Answer:

B. $70^{\circ}$