Question

1. lines ab and bc are perpendicular. the dashed rays bisect angles abd and cbd. explain why the measure of angle ebf is 45 degrees.

Explanation:

Step1: Determine \( \angle ABC \)

Since \( AB \perp BC \), by the definition of perpendicular lines, \( \angle ABC = 90^\circ \). This means \( \angle ABD+\angle CBD=\angle ABC = 90^\circ \).

Step2: Use the angle - bisector definition

Let \( BE \) bisect \( \angle ABD \) and \( BF \) bisect \( \angle CBD \). By the definition of an angle bisector, \( \angle EBD=\frac{1}{2}\angle ABD \) and \( \angle FBD=\frac{1}{2}\angle CBD \).

Step3: Calculate \( \angle EBF \)

We know that \( \angle EBF=\angle EBD + \angle FBD \). Substitute \( \angle EBD=\frac{1}{2}\angle ABD \) and \( \angle FBD=\frac{1}{2}\angle CBD \) into the formula for \( \angle EBF \):
\( \angle EBF=\frac{1}{2}\angle ABD+\frac{1}{2}\angle CBD=\frac{1}{2}(\angle ABD + \angle CBD) \)
Since \( \angle ABD+\angle CBD = 90^\circ \), then \( \angle EBF=\frac{1}{2}\times90^\circ = 45^\circ \).

Answer:

The measure of angle \( EBF \) is \( 45^\circ \).