We are given a circle with an angle $VYX$ formed by two secants intersecting outside the circle. The measures of the intercepted arcs $VW$ and $VX$ are given as $m \stackrel{\frown}{VW} = 140^\circ$ and $m \stackrel{\frown}{VX} = 62^\circ$. We need to find the measure of angle $VYX$.

1. Problem Description

We are given a circle with an angle $VYX$ formed by two secants intersecting outside the circle. The measures of the intercepted arcs $VW$ and $VX$ are given as $m \stackrel{\frown}{VW} = 140^\circ$ and $m \stackrel{\frown}{VX} = 62^\circ$. We need to find the measure of angle $VYX$.

2. Solution Steps

The measure of an angle formed by two secants intersecting outside a circle is equal to one-half the positive difference of the measures of the intercepted arcs.

In this case, the intercepted arcs are $VW$ and $VX$.

$$m\angle VYX = \frac{1}{2} (m \stackrel{\frown}{VW} - m \stackrel{\frown}{VX})$$

Substitute the given values:

$$m\angle VYX = \frac{1}{2} (140^\circ - 62^\circ)$$

$$m\angle VYX = \frac{1}{2} (78^\circ)$$

$$m\angle VYX = 39^\circ$$

3. Final Answer

The measure of angle $VYX$ is $39^\circ$.

$m \angle VYX = 39^\circ$