The problem asks us to find the measure of arc $ST$ given that $m \angle PUR = 31^\circ$ and $m \stackrel{\frown}{PR} = 99^\circ$.

1. Problem Description

The problem asks us to find the measure of arc $ST$ given that $m \angle PUR = 31^\circ$ and $m \stackrel{\frown}{PR} = 99^\circ$.

2. Solution Steps

The measure of an angle formed by two secants intersecting outside a circle is half the difference of the intercepted arcs. In this case, $\angle PUR$ is formed by secants $\overline{UR}$ and $\overline{UP}$, intersecting outside the circle. The intercepted arcs are $\stackrel{\frown}{PR}$ and $\stackrel{\frown}{ST}$. Therefore,

$$m \angle PUR = \frac{1}{2} |m \stackrel{\frown}{PR} - m \stackrel{\frown}{ST}|$$

We are given that $m \angle PUR = 31^\circ$ and $m \stackrel{\frown}{PR} = 99^\circ$. Substituting these values into the formula, we get

$$31 = \frac{1}{2} |99 - m \stackrel{\frown}{ST}|$$

Multiply both sides by 2:

$$62 = |99 - m \stackrel{\frown}{ST}|$$

We have two cases:

Case 1: $99 - m \stackrel{\frown}{ST} = 62$

$m \stackrel{\frown}{ST} = 99 - 62 = 37$

Case 2: $99 - m \stackrel{\frown}{ST} = -62$

$m \stackrel{\frown}{ST} = 99 + 62 = 161$

Since arcs are generally less than or equal to $180^\circ$, we will use $37^\circ$.

3. Final Answer

37