Question

Sam constructed the angle bisector of ∠
ABC by constructing BD−→−
. Sam then constructed the angle bisector of ∠
DBC by constructing BE−→−
. If the measurement of ∠
ABC is 54 degrees, what is the measurement of ∠
EBC?(1 point)
Responses

The measurement of ∠
EBC is 14 degrees.
The measurement of angleEBC is 14 degrees.

The measurement of ∠
EBC is 27 degrees.
The measurement of angleEBC is 27 degrees.

The measurement of ∠
EBC is 108 degrees.
The measurement of angleEBC is 108 degrees.

The measurement of ∠
EBC is 13.5 degrees.

Answer 1

To find the measurement of angle \( \angle EBC \), we will analyze the angle bisectors that Sam constructed.

1. **First, consider \(\angle ABC\)**:
- It measures \( 54^\circ \).
- Sam constructs the angle bisector \( \overline{BD} \), which divides \( \angle ABC \) into two equal parts.
- Therefore, each part measures:
\[
\frac{54^\circ}{2} = 27^\circ
\]
So:
\[
\angle ABD = 27^\circ \quad \text{and} \quad \angle DBC = 27^\circ
\]

2. **Next, consider \(\angle DBC\)**:
- Sam now constructs the angle bisector \( \overline{BE} \) for \(\angle DBC\), which also measures \( 27^\circ \).
- Again, the angle bisector divides \( \angle DBC \) into two equal parts:
\[
\frac{27^\circ}{2} = 13.5^\circ
\]
So:
\[
\angle EBD = 13.5^\circ \quad \text{and} \quad \angle EBC = 13.5^\circ
\]

Thus, the measurement of \( \angle EBC \) is:\[\boxed{13.5 \text{ degrees}}\]