Since triangles \( ABC \) and \( DEF \) are similar (denoted as \( ABC \sim DEF \)), the corresponding angles are equal. This means:
\[ m<A = m<D, \quad m<B = m<E, \quad m<C = m<F \]
We are given that \( m<A = 62^\circ \) and \( m<E = 13^\circ \).
Since \( m<B \) corresponds to \( m<E \), we can find \( m<B \): \[ m<B = m<E = 13^\circ \]
Now, we can use the fact that the sum of the angles in a triangle is \( 180^\circ \) to find \( m<C \): \[ m<A + m<B + m<C = 180^\circ \] \[ 62^\circ + 13^\circ + m<C = 180^\circ \]
Calculating the left side: \[ 75^\circ + m<C = 180^\circ \]
Now, solving for \( m<C \): \[ m<C = 180^\circ - 75^\circ = 105^\circ \]
Thus, the measure of angle \( C \) is: \[ \boxed{105^\circ} \]
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