Since triangles \( \triangle ABC \) and \( \triangle DEF \) are similar (denoted as \( \triangle ABC \sim \triangle DEF \)), they have corresponding angles that are equal.
We are given:
- \( m\angle A = 52^\circ \)
- \( m\angle E = 11^\circ \)
The sum of the angles in a triangle is always \( 180^\circ \). Therefore, we can find the measures of angles B and D in triangles \( ABC \) and \( DEF \) respectively.
First, we find \( m\angle B \) using the property of triangles: \[ m\angle B = 180^\circ - m\angle A - m\angle C \] For triangle \( DEF \), we can do similarly: \[ m\angle D = 180^\circ - m\angle E - m\angle F \]
Since \( m\angle B = m\angle D \) and \( m\angle A = m\angle E \):
Let’s find \( m\angle C \) (which corresponds to \( m\angle F \)):
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From triangle ABC: \[ m\angle A + m\angle B + m\angle C = 180^\circ \] So, we have: \[ 52^\circ + m\angle B + m\angle C = 180^\circ \tag{1} \]
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From triangle DEF: \[ m\angle E + m\angle D + m\angle F = 180^\circ \] So, we have: \[ 11^\circ + m\angle D + m\angle C = 180^\circ \tag{2} \]
Now, since \( m\angle C = m\angle F \): Substituting \( m\angle D \) with \( m\angle B \) gives: \[ m\angle D = 180^\circ - 11^\circ - m\angle C \]
Using the fact that triangles are similar, we can express \( m\angle A + m\angle B + m\angle C = 180^\circ \) from triangle ABC and \( m\angle E + m\angle D + m\angle F = 180^\circ \) from triangle DEF.
Solving equation: \[ 52^\circ + m\angle D + m\angle C = 180^\circ \] Using that \( m\angle D = 180^\circ - 11^\circ - m\angle C \): \[ 52^\circ + (180^\circ - 11^\circ - m\angle C) + m\angle C = 180^\circ \]
Simplifying this gives: \[ 52^\circ + 180^\circ - 11^\circ = 180^\circ \] \[ 221^\circ - m\angle C = 180^\circ \] \[ m\angle C = 221^\circ - 180^\circ = 41^\circ \]
Thus, the measure of angle \( C \) is: \[ \boxed{41^\circ} \]
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