Since triangles \( \triangle ABC \) and \( \triangle DEF \) are similar, the corresponding angles of the triangles are equal.
We know:
- \( m\angle A = 52^\circ \)
- \( m\angle E = 11^\circ \)
Using the property of similar triangles, we can write: \[ m\angle A + m\angle B + m\angle C = 180^\circ \] \[ m\angle D + m\angle E + m\angle F = 180^\circ \]
Since the angles in both triangles add up to \( 180^\circ \), we can calculate \( m\angle B \) in \( \triangle ABC \):
\[ m\angle B = 180^\circ - m\angle A - m\angle C \]
Similarly, we can express \( m\angle F \) in \( \triangle DEF \):
\[ m\angle F = 180^\circ - m\angle D - m\angle E \]
Since the triangles are similar, the angles correspond as follows:
- \( m\angle A = m\angle D \)
- \( m\angle B = m\angle E \)
- \( m\angle C = m\angle F \)
From this, we see that: \[ m\angle A = m\angle D = 52^\circ \]
Now, we can calculate \( m\angle B \):
\[ m\angle B = m\angle E = 11^\circ \]
Putting all this into the triangle angle sum for triangle \( ABC \):
\[ m\angle A + m\angle B + m\angle C = 180^\circ \] \[ 52^\circ + 11^\circ + m\angle C = 180^\circ \] \[ 63^\circ + m\angle C = 180^\circ \] \[ m\angle C = 180^\circ - 63^\circ = 117^\circ \]
Thus, the measure of angle \( C \) is:
\[ \boxed{117^\circ} \]