Given that triangles \( \triangle ABC \sim \triangle DEF \), we know that corresponding angles of similar triangles are equal. Thus, we can write:
\[ m\angle A = m\angle D \] \[ m\angle B = m\angle E \] \[ m\angle C = m\angle F \]
We know:
- \( m\angle A = 52^\circ \)
- \( m\angle E = 11^\circ \)
Since \( m\angle B = m\angle E \), we have:
\[ m\angle B = 11^\circ \]
To find \( m\angle C \) (which corresponds to \( m\angle F \)), we first find \( m\angle D \):
Using the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[ m\angle A + m\angle B + m\angle C = 180^\circ \]
Substituting the known values:
\[ 52^\circ + 11^\circ + m\angle C = 180^\circ \]
This simplifies to:
\[ 63^\circ + m\angle C = 180^\circ \]
Now, solving for \( m\angle C \):
\[ m\angle C = 180^\circ - 63^\circ = 117^\circ \]
Thus, the measure of angle \( C \) is:
\[ \boxed{117^\circ} \]
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