In the figure \( \triangle A B C \) is rotated clockwise and counterclockwise. Therefore, \( m \angle D B A= \) and \( m \angle E B C= \)

1. A rotation about a fixed point sends every ray emerging from that point to a new ray making an equal‐angle turn. In other words, if a ray (for example, \(\overrightarrow{B A}\)) is rotated about \(B\) by an angle \(\theta\), then the angle between \(\overrightarrow{B A}\) and its rotated image is \(\theta\). 2. In the figure the triangle \(\triangle A B C\) is rotated about \(B\) in two different ways. In the clockwise rotation the ray \(\overrightarrow{B A}\) is carried onto \(\overrightarrow{B D}\). Therefore, the angle between \(\overrightarrow{B A}\) and \(\overrightarrow{B D}\) is the measure of the clockwise rotation. That is, \[ m\angle D B A=\theta \] where \(\theta\) is the measure of the clockwise rotation. 3. Similarly, in the counterclockwise rotation the ray \(\overrightarrow{B C}\) is carried onto \(\overrightarrow{B E}\). Thus, the angle between \(\overrightarrow{B C}\) and its image \(\overrightarrow{B E}\) is the measure of the counterclockwise rotation. That is, \[ m\angle E B C=\theta \] where (possibly the same or a different) \(\theta\) is the measure of the counterclockwise rotation. Without any additional numerical information given in the diagram, the answer is that each of the angles is equal to the corresponding rotation angle.