Question

a student uses a compass and straightedge to construct the bisector of ∠abc as shown. indicate whether each statement is true. the measure of ∠dbc is equal to half the measure of ∠abc. select choice the measure of ∠abd is equal to the measure of ∠abc. select choice the measure of ∠abd is equal to the measure of ∠dbc. select choice

Explanation:

Step1: Recall angle - bisector definition

An angle - bisector divides an angle into two equal parts. Since $\overrightarrow{BD}$ is the bisector of $\angle ABC$, by definition, $\angle ABD=\angle DBC$ and $\angle DBC=\frac{1}{2}\angle ABC$, $\angle ABD = \frac{1}{2}\angle ABC$.

Step2: Analyze the first statement

The measure of $\angle DBC$ is equal to half the measure of $\angle ABC$. Since $\overrightarrow{BD}$ bisects $\angle ABC$, this statement is True.

Step3: Analyze the second statement

The measure of $\angle ABD$ is equal to the measure of $\angle ABC$. Since $\angle ABD=\frac{1}{2}\angle ABC$, this statement is False.

Step4: Analyze the third statement

The measure of $\angle ABD$ is equal to the measure of $\angle DBC$. Because $\overrightarrow{BD}$ is the bisector of $\angle ABC$, this statement is True.

Answer:

The measure of $\angle DBC$ is equal to half the measure of $\angle ABC$: True
The measure of $\angle ABD$ is equal to the measure of $\angle ABC$: False
The measure of $\angle ABD$ is equal to the measure of $\angle DBC$: True