Since triangles \( \triangle CDE \) and \( \triangle PQR \) are similar (\( \triangle CDE \sim \triangle PQR \)), their corresponding angles are equal.
Given:
- \( m\angle C = 13^\circ \)
- \( m\angle D = 65^\circ \)
To find the measure of angle \( E \) in triangle \( CDE \), we can use the fact that the sum of the angles in a triangle is \( 180^\circ \):
\[ m\angle E = 180^\circ - m\angle C - m\angle D \] \[ m\angle E = 180^\circ - 13^\circ - 65^\circ \] \[ m\angle E = 180^\circ - 78^\circ = 102^\circ \]
Now that we have all the angles in triangle \( CDE \):
- \( m\angle C = 13^\circ \)
- \( m\angle D = 65^\circ \)
- \( m\angle E = 102^\circ \)
Using the fact that corresponding angles in similar triangles are equal, we can now identify the angles in triangle \( PQR \):
- \( m\angle P = m\angle C = 13^\circ \)
- \( m\angle Q = m\angle D = 65^\circ \)
- \( m\angle R = m\angle E = 102^\circ \)
Thus:
- \( m\angle P = 13^\circ \)
- \( m\angle Q = 65^\circ \)
- \( m\angle R = 102^\circ \)
So the measures of angles \( P \), \( Q \), and \( R \) are:
- \( m\angle P = 13^\circ \)
- \( m\angle Q = 65^\circ \)
- \( m\angle R = 102^\circ \)