Since triangles \( \triangle ABC \) and \( \triangle DEF \) are similar, the corresponding angles are equal. Therefore, we can use the information about the angles to find the measure of \( \angle C \).
In triangle \( ABC \):
\[ m\angle A + m\angle B + m\angle C = 180^\circ \]
Given:
- \( m\angle A = 52^\circ \)
Let \( m\angle B = m\angle E = 11^\circ \) (because corresponding angles in similar triangles are equal).
Now we can find \( m\angle C \):
\[ 52^\circ + 11^\circ + m\angle C = 180^\circ \]
\[ 63^\circ + m\angle C = 180^\circ \]
\[ m\angle C = 180^\circ - 63^\circ \]
\[ m\angle C = 117^\circ \]
Thus, the measure of angle \( C \) is:
\[ \boxed{117^\circ} \]
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