Since triangles \( \triangle CDE \) and \( \triangle PQR \) are similar (denoted as \( \triangle CDE \sim \triangle PQR \)), the corresponding angles are equal.
Given:
- \( m\angle C = 13^\circ \)
- \( m\angle D = 65^\circ \)
To find the measure of angle \( E \) in triangle \( CDE \):
\[ m\angle E = 180^\circ - m\angle C - m\angle D \] \[ m\angle E = 180^\circ - 13^\circ - 65^\circ = 180^\circ - 78^\circ = 102^\circ \]
Now, we find the corresponding angles in triangle \( PQR \):
- Since \( m\angle C = m\angle P \), we have \( m\angle P = 13^\circ \).
- Since \( m\angle D = m\angle Q \), we have \( m\angle Q = 65^\circ \).
- Since \( m\angle E = m\angle R \), we have \( m\angle R = 102^\circ \).
Thus, the measures of the angles in triangle \( PQR \) are:
- \( m\angle P = 13^\circ \)
- \( m\angle Q = 65^\circ \)
- \( m\angle R = 102^\circ \)
Final answer:
- \( m\angle P = 13^\circ \)
- \( m\angle Q = 65^\circ \)
- \( m\angle R = 102^\circ \)