Final answer:
The measure of angle EBF is 45 degrees because it is created by bisecting a 90-degree angle formed by perpendicular lines AB and BC; hence, each bisected angle measures 45 degrees.
Step-by-step explanation:
The student's question relates to angle measurements in geometric figures involving perpendicular lines and angle bisectors. Since lines AB and BC are perpendicular, they form a 90-degree angle at point B. The dashed rays are given to bisect angles ABD and CBD, which implies that each angle is divided into two equal parts. If the sum of angles ABD and CBD is 90 degrees (due to the perpendicular lines), each bisected angle (ABE and CBE) must measure 45 degrees. This is because the bisected angles are congruent (they have the same measure).
The angle EBF is formed by the intersection of the two dashed rays, which are bisectors of angles ABE and CBE respectively. Therefore, angle EBF is also 45 degrees since it shares angle EBE with angle ABE, which we established as 45 degrees. The reasoning involves the understanding that a bisected right angle creates two angles of 45 degrees each.