Question

point b is the midpoint of \overline{ac}. which statements about the figure must be true? select three options. \angle dbc is bisected by ray bd. \angle abc is bisected by ray bd. bc = \frac{1}{2}ac \overline{db} \cong \overline{bc} 2m\angle dbc = m\angle abc

Explanation:

Step1: Analyze the angle - bisector concept

A ray bisects an angle if it divides the angle into two equal parts. Ray \(BD\) divides \(\angle ABC\) into \(\angle ABD\) and \(\angle DBC\). Since \(\angle ABD = \angle DBC=90^{\circ}\), \(\angle ABC\) is bisected by ray \(BD\).

Step2: Use the mid - point property

If \(B\) is the mid - point of \(\overline{AC}\), then by the definition of a mid - point, \(BC=\frac{1}{2}AC\) (because \(AB = BC\) and \(AC=AB + BC\)).

Step3: Analyze the angle - measure relationship

Since \(\angle ABC = 180^{\circ}\) (a straight angle) is wrong. Wait, no, \(\angle ABC\) is composed of two right angles (\(\angle ABD\) and \(\angle DBC\)). \(\angle ABC=\angle ABD+\angle DBC\), and since \(\angle ABD=\angle DBC = 90^{\circ}\), \(2m\angle DBC=m\angle ABC\) (because \(m\angle ABC=m\angle ABD + m\angle DBC\) and \(m\angle ABD=m\angle DBC\)).

• For \(\angle DBC\) is bisected by ray \(BD\): A ray cannot bisect itself. So this is false.
• For \(\overline{DB}\cong\overline{BC}\): There is no information (from the mid - point property of \(B\) on \(\overline{AC}\) and the right - angle information) to suggest that the lengths of \(DB\) and \(BC\) are equal. So this is false.

Answer:

\(\angle ABC\) is bisected by ray \(BD\), \(BC = \frac{1}{2}AC\), \(2m\angle DBC=m\angle ABC\)