Since the triangles \( \triangle CDE \sim \triangle PQR \) are similar, the corresponding angles are equal.
Given:
- \( m\angle C = 13^\circ \)
- \( m\angle D = 65^\circ \)
First, we can calculate the measure of angle \( E \) in triangle \( CDE \):
\[ m\angle E = 180^\circ - m\angle C - m\angle D \] \[ m\angle E = 180^\circ - 13^\circ - 65^\circ \] \[ m\angle E = 180^\circ - 78^\circ = 102^\circ \]
Now, we know the measures of angles in triangle \( CDE \):
- \( m\angle C = 13^\circ \)
- \( m\angle D = 65^\circ \)
- \( m\angle E = 102^\circ \)
Because the triangles are similar, the corresponding angles in triangle \( PQR \) will have the same measures:
- Let \( m\angle P = m\angle C = 13^\circ \)
- Let \( m\angle Q = m\angle D = 65^\circ \)
- Let \( m\angle R = m\angle E = 102^\circ \)
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:
\[ m\angle P = 13^\circ, \quad m\angle Q = 65^\circ, \quad m\angle R = 102^\circ \]