Since triangles \( \triangle ABC \) and \( \triangle DEF \) are similar, the corresponding angles are equal. This means the angles in triangle \( ABC \) correspond to the angles in triangle \( DEF \).
We know:
- \( m\angle A = 52^\circ \)
- \( m\angle E = 11^\circ \)
To find \( m\angle C \), we need to first find \( m\angle B \). The sum of the angles in triangle \( ABC \) is \( 180^\circ \).
Using the property of similar triangles:
\[ m\angle B + m\angle C + m\angle A = 180^\circ \]
This can also be represented in triangle \( DEF \):
\[ m\angle D + m\angle E + m\angle F = 180^\circ \]
Since \( m\angle E \) corresponds to \( m\angle B \), we can find \( m\angle B \):
\[ m\angle B = m\angle E = 11^\circ \]
Now we can find \( m\angle C \):
\[ m\angle A + m\angle B + m\angle C = 180^\circ \]
Substituting in the known values:
\[ 52^\circ + 11^\circ + m\angle C = 180^\circ \]
Calculating \( 52^\circ + 11^\circ \):
\[ 63^\circ + m\angle C = 180^\circ \]
Now, subtract \( 63^\circ \) from both sides:
\[ m\angle C = 180^\circ - 63^\circ = 117^\circ \]
Therefore, the measure of angle \( C \) is:
\[ \boxed{117^\circ} \]
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