Question

3 from unit 1, lesson 19
lines ab and bc are perpendicular. the dashed rays bisect angles abd and cbd.
explain why the measure of angle ebf is 45°.

Explanation:

Step1: Determine the measure of ∠ABC

Since \(AB\) and \(BC\) are perpendicular, \(\angle ABC = 90^{\circ}\).

Step2: Use the angle - bisector property

Let \(\angle ABD=x\) and \(\angle CBD = y\), so \(x + y=90^{\circ}\).
The dashed ray \(BE\) bisects \(\angle ABD\), so \(\angle EBD=\frac{x}{2}\).
The dashed ray \(BF\) bisects \(\angle CBD\), so \(\angle FBD=\frac{y}{2}\).

Step3: Calculate the measure of \(\angle EBF\)

\(\angle EBF=\angle EBD+\angle FBD\).
Substitute \(\angle EBD=\frac{x}{2}\) and \(\angle FBD=\frac{y}{2}\) into the formula: \(\angle EBF=\frac{x + y}{2}\).
Since \(x + y = 90^{\circ}\), then \(\angle EBF=\frac{90^{\circ}}{2}=45^{\circ}\).

Answer:

The measure of \(\angle EBF\) is \(45^{\circ}\) because \(\angle ABC = 90^{\circ}\), and using the angle - bisector property \(\angle EBF=\frac{1}{2}(\angle ABD+\angle CBD)=\frac{1}{2}\times90^{\circ}=45^{\circ}\).