Since triangle ABC ~ triangle DEF, the corresponding angles of the triangles are equal. Therefore, the measures of angles in triangle ABC correspond to the measures of angles in triangle DEF.
Given:
- \( m\angle A = 52^\circ \)
- \( m\angle E = 11^\circ \)
To find \( m\angle C \), we first need to determine the measure of angle \( B \) in triangle ABC:
The sum of angles in a triangle is always 180 degrees, so we can express the angles in triangle ABC as follows:
\[ m\angle A + m\angle B + m\angle C = 180^\circ \]
To find \( m\angle B \), we need to find the angles \( D \) and \( F \) in triangle DEF that correspond to angles \( B \) and \( C \) in triangle ABC, respectively.
Since the triangles are similar (ABC ~ DEF), we can set up the equation based on corresponding angles:
\[ m\angle B = m\angle D = 180^\circ - (m\angle A + m\angle E) \]
Substituting the known values:
\[ m\angle B = 180^\circ - (52^\circ + 11^\circ) \] \[ m\angle B = 180^\circ - 63^\circ = 117^\circ \]
Now we substitute \( m\angle A \) and \( m\angle B \) back into the triangle sum for triangle ABC to find \( m\angle C \):
\[ m\angle C = 180^\circ - (m\angle A + m\angle B) \] \[ m\angle C = 180^\circ - (52^\circ + 117^\circ) \] \[ m\angle C = 180^\circ - 169^\circ = 11^\circ \]
Thus, the measure of angle \( C \) is \( 11^\circ \).