To solve for the measures of angles \( P \), \( Q \), and \( R \) in triangles \( \Delta CDE \) and \( \Delta PQR \), we use the properties of similar triangles.
We know the following about \( \Delta CDE \):
- \( m \angle C = 13^\circ \)
- \( m \angle D = 65^\circ \)
First, we need to find the measure of angle \( E \) in \( \Delta CDE \): \[ m \angle E = 180^\circ - m \angle C - m \angle D = 180^\circ - 13^\circ - 65^\circ = 180^\circ - 78^\circ = 102^\circ \]
Now we have the angles in \( \Delta CDE \):
- \( m \angle C = 13^\circ \)
- \( m \angle D = 65^\circ \)
- \( m \angle E = 102^\circ \)
Since \( \Delta CDE \) and \( \Delta PQR \) are similar triangles, the corresponding angles of the two triangles are equal.
Let’s assign angles of \( \Delta PQR \):
- \( m \angle P \) corresponds to \( m \angle C \)
- \( m \angle Q \) corresponds to \( m \angle D \)
- \( m \angle R \) corresponds to \( m \angle E \)
Thus, the measures of angles in triangle \( \Delta PQR \) will be:
- \( m \angle P = m \angle C = 13^\circ \)
- \( m \angle Q = m \angle D = 65^\circ \)
- \( m \angle R = m \angle E = 102^\circ \)
Final answers:
- \( m<P = 13^\circ \)
- \( m<Q = 65^\circ \)
- \( m<R = 102^\circ \)