This problem involves finding the measure of angle $\angle S$, which is formed by two intersecting secants in a circle.

The formula to calculate the angle formed by two secants that intersect outside a circle is:

$m\angle S = \frac{1}{2} \left( m\overset{\frown}{MP} - m\overset{\frown}{FG} \right)$

Here:

• $m\overset{\frown}{MP} = 95^\circ$
• $m\overset{\frown}{FG} = 19^\circ$

Substituting these values into the formula:

$m\angle S = \frac{1}{2} \left( 95^\circ - 19^\circ \right) = \frac{1}{2} \times 76^\circ = 38^\circ$

Thus, the measure of $\angle S$ is 38°.

Would you like more details or explanations? Here are 5 related questions to deepen your understanding:

1. How is the angle formed by two secants outside a circle related to the intercepted arcs?
2. What happens if both secants were tangent to the circle at their endpoints?
3. Can this formula be applied when the secants intersect inside the circle?
4. What if the arcs $MP$ and $FG$ were given as degrees in radians? How would the calculation change?
5. How would you find $\angle S$ if the secants intersected inside the circle?

Tip: The key to solving problems with intersecting secants is to always focus on the relationship between the angle and the intercepted arcs!