To find the measure of angle \( XWY \) (angle \( W \)), we can use the property of exterior angles in a triangle. The exterior angle is equal to the sum of the two opposite interior angles.
In this case, the exterior angle to angle \( W \) is given as \( 74^\circ \). Therefore, we can write:
\[ \text{Exterior angle} = \text{Angle } X + \text{Angle } Y \]
From the problem, we know that:
\[ \text{Exterior angle} = 74^\circ \]
If we let \( \angle W = \text{Angle } W \), which we want to find, then the interior angles \( X \) and \( Y \) in triangle \( XYW \) also satisfy:
\[ \angle X + \angle Y + \angle W = 180^\circ \]
From the exterior angle property, we can relate \( \angle W \) to \( 74^\circ \):
\[ \angle X + \angle Y = 74^\circ \]
Now substituting this into the triangle angle sum formula, we get:
\[ 74^\circ + \angle W = 180^\circ \]
To find \( \angle W \):
\[ \angle W = 180^\circ - 74^\circ = 106^\circ \]
Thus, the measure of angle \( XWY \) (or \( \angle W \)) is:
\[ \boxed{106 \text{ degrees}} \]