Since triangles \( \triangle ABC \sim \triangle DEF \) are similar, the corresponding angles are equal.
We know:
- \( m\angle A = 52^\circ \)
- \( m\angle E = 11^\circ \)
To find \( m\angle C \), we can use the fact that the sum of the angles in a triangle is always \( 180^\circ \).
First, we can find \( m\angle B \) using the properties of similar triangles:
Since the corresponding angles in similar triangles are equal, we have: \[ m\angle B = m\angle D \] and \[ m\angle C = m\angle F \]
We can find \( m\angle C \) using the fact that the sum of the angles in triangle \( ABC \) equals \( 180^\circ \): \[ m\angle A + m\angle B + m\angle C = 180^\circ \]
Since \( m\angle D + m\angle E + m\angle F = 180^\circ \) and we know \( m\angle E = 11^\circ \): \[ m\angle D + 11^\circ + m\angle F = 180^\circ \] Rearranging gives: \[ m\angle D + m\angle F = 169^\circ \]
Thus, we can express: \[ m\angle B + m\angle C = 169^\circ \]
Now, since \( m\angle A = 52^\circ \), we have: \[ 52^\circ + m\angle B + m\angle C = 180^\circ \]
Combining the two equations:
- \( m\angle B + m\angle C = 169^\circ \)
- \( m\angle B + m\angle C + 52^\circ = 180^\circ \)
From the first equation, we have: \[ m\angle C = 169^\circ - m\angle B \]
Substituting this into the second equation: \[ 52^\circ + m\angle B + (169^\circ - m\angle B) = 180^\circ \] \[ 52^\circ + 169^\circ = 180^\circ \] \[ 221^\circ = 180^\circ \]
This shows that there's a contradiction if we solve these equations separately, thus returning to: From the corresponding angles, we also observe:
\[ m\angle C + m\angle F = m\angle A + m\angle E = 52^\circ + 11^\circ = 63^\circ \]
Thus, it confirms that \( m\angle C = 63^\circ - m\angle F \) but given \( D + E + F = 180^\circ \), leading to the relevant identity we can shuffle between these angles; thus finally the result without getting losses would be:
\[ m\angle C \approx 180^\circ - (52^\circ + 11^\circ) = 117^\circ. \]
Hence, the final calculation gives that \( m\angle C \) equals
\[ \boxed{117^\circ}. \]