We need to find the measure of angle $R$ (denoted as $m\angle R$) given a circle with tangents intersecting at point $R$. We are given the measures of intercepted arcs $PQ = 40^\circ$ and $VT = 120^\circ$.

1. Problem Description

We need to find the measure of angle $R$ (denoted as $m\angle R$) given a circle with tangents intersecting at point $R$. We are given the measures of intercepted arcs $PQ = 40^\circ$ and $VT = 120^\circ$.

2. Solution Steps

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

In this case, the intercepted arcs are arc $VT$ and arc $PQ$.

$$m\angle R = \frac{1}{2} |m(\text{arc } VT) - m(\text{arc } PQ)|$$

Substituting the given values:

$$m\angle R = \frac{1}{2} |120^\circ - 40^\circ|$$

$$m\angle R = \frac{1}{2} |80^\circ|$$

$$m\angle R = \frac{1}{2} (80^\circ)$$

$$m\angle R = 40^\circ$$

3. Final Answer

40