Question

in the figure, the measure of (widehat{rus}) is 222°. what is the measure of (angle tsr?) enter your answer in the box.

Explanation:

Step1: Recall the tangent - secant angle formula

The measure of an angle formed by a tangent and a secant is half the difference of the measures of the intercepted arcs. The whole circle is 360°. So the measure of arc $\overset{\frown}{RS}=360^{\circ}- 222^{\circ}=138^{\circ}$.

Step2: Apply the formula for $\angle TSR$

The measure of $\angle TSR=\frac{1}{2}(\text{measure of arc}\ \overset{\frown}{RU}-\text{measure of arc}\ \overset{\frown}{RS})$. Substitute the values: $\text{measure of arc}\ \overset{\frown}{RU} = 222^{\circ}$ and $\text{measure of arc}\ \overset{\frown}{RS}=138^{\circ}$. Then $\angle TSR=\frac{1}{2}(222 - 138)$.

Step3: Calculate the value of $\angle TSR$

First, calculate $222-138 = 84$. Then $\frac{1}{2}\times84 = 42$.

Answer:

$42$