Since triangles \( \triangle CDE \) and \( \triangle PQR \) are similar (denoted as \( \triangle CDE \sim \triangle PQR \)), their corresponding angles are equal.
Given:
- \( m\angle C = 13^\circ \)
- \( m\angle D = 65^\circ \)
To find \( m\angle E \): Using the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[ m\angle E = 180^\circ - m\angle C - m\angle D \] \[ m\angle E = 180^\circ - 13^\circ - 65^\circ \] \[ m\angle E = 102^\circ \]
Now we have the angles of triangle \( CDE \):
- \( m\angle C = 13^\circ \)
- \( m\angle D = 65^\circ \)
- \( m\angle E = 102^\circ \)
Since the triangles are similar, we can assign the angles of \( \triangle PQR \) based on the corresponding angles of \( \triangle CDE \). We can let:
- \( m\angle P = m\angle C = 13^\circ \)
- \( m\angle Q = m\angle D = 65^\circ \)
- \( m\angle R = m\angle E = 102^\circ \)
Thus, the measures of angles \( P \), \( Q \), and \( R \) are:
- \( m\angle P = 13^\circ \)
- \( m\angle Q = 65^\circ \)
- \( m\angle R = 102^\circ \)