Question

what is $moverset{\frown}{su}$?
○ $49^circ$
○ $77^circ$
○ $98^circ$
○ $161^circ$
the circle has points r, s on a diameter, point u on the circle, and an external point t. the measure of arc ru is $119^circ$, and the measure of angle stu is $21^circ$.

Explanation:

Step1: Recall secant-secant angle formula

The measure of an angle formed by two secants outside a circle is half the difference of the measures of the intercepted arcs. The formula is:
$$m\angle T = \frac{1}{2}(m\overset{\frown}{RU} - m\overset{\frown}{SU})$$

Step2: Substitute known values

We know $m\angle T = 21^\circ$ and $m\overset{\frown}{RU} = 119^\circ$. Plug these into the formula:
$$21^\circ = \frac{1}{2}(119^\circ - m\overset{\frown}{SU})$$

Step3: Solve for $m\overset{\frown}{SU}$

Multiply both sides by 2:
$$42^\circ = 119^\circ - m\overset{\frown}{SU}$$
Rearrange to isolate $m\overset{\frown}{SU}$:
$$m\overset{\frown}{SU} = 119^\circ - 42^\circ$$
$$m\overset{\frown}{SU} = 77^\circ$$

Answer:

77° (Option B)