In a rhombus, all sides are equal, and the opposite angles are equal. Additionally, the diagonals of a rhombus bisect each other at right angles.
Given that \( m\angle WVX = 60^\circ \) is one of the angles at the vertices of the rhombus, we can find the measure of other angles.
Since \( WVXY \) is a rhombus, we have:
- Opposite angles are equal.
- Adjacent angles are supplementary (add up to \( 180^\circ \)).
Since we are given that \( m\angle WVX = 60^\circ \), the adjacent angle \( m\angle XYW \) would be:
\[ m\angle XYW = 180^\circ - m\angle WVX = 180^\circ - 60^\circ = 120^\circ. \]
The opposite angle, \( m\angle XYW \), would also be \( 60^\circ\).
Finally, \( m\angle WVX \) is already given as \( 60^\circ\).
Thus, we can conclude:
\[ m\angle WVX = 60^\circ. \]
So, the final answer is \( m\angle WVX = 60^\circ \).