In similar triangles, the corresponding angles are equal. Since \( \triangle ABC \sim \triangle DEF \), the corresponding angles \( A \) and \( E \) are related. We know:
- \( \text{Measure of Angle A} = 52^\circ \)
- \( \text{Measure of Angle E} = 11^\circ \)
To find the measure of angle \( C \) in triangle \( ABC \), we first need to determine angle \( F \) in triangle \( DEF \). In a triangle, the sum of the angles is always \( 180^\circ \). Thus, we can find angle \( F \) as follows:
\[ \text{Measure of Angle F} = 180^\circ - \text{Measure of Angle E} - \text{Measure of Angle D} \]
Since we do not have the value of angle \( D \), we can't find angle \( F \) directly. However, we can find angle \( C \) by recognizing the relationship between the angles.
We know that in similar triangles:
\[ \text{Measure of Angle A} + \text{Measure of Angle B} + \text{Measure of Angle C} = 180^\circ \]
Since we already have \( \text{Measure of Angle A} = 52^\circ \), we can express it as:
\[ 52^\circ + \text{Measure of Angle B} + C = 180^\circ \]
Let's assume that Angle \( B \) corresponds to angle \( E \):
\[ \text{Measure of Angle B} = \text{Measure of Angle E} = 11^\circ \]
Now, substituting this into the equation gives:
\[ 52^\circ + 11^\circ + C = 180^\circ \]
Now, simplifying this:
\[ 63^\circ + C = 180^\circ \]
Thus:
\[ C = 180^\circ - 63^\circ = 117^\circ \]
So, the measure of angle \( C \) is \( 117^\circ \).
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