Question

given that \\( \overrightarrow { ba } \\) bisects \\( \angle dbc \\), which statement must be true?
\\( \bigcirc m \angle abd = m \angle abc \\)
\\( \bigcirc \overline { a b } \cong \overline { b c } \\)
\\( \bigcirc b \\) is the midpoint of \\( \overline { d c } \\).
\\( \bigcirc m \angle dbc = 90 ^ { \circ } \\)

Explanation:

Step1: Recall the definition of an angle bisector

An angle bisector divides an angle into two equal - measure angles. If \(\overrightarrow{BA}\) bisects \(\angle DBC\), then \(\angle ABD\) and \(\angle ABC\) are the two angles formed by the bisector \(\overrightarrow{BA}\) of \(\angle DBC\).
By the definition of an angle bisector, \(m\angle ABD=m\angle ABC\).

Step2: Analyze other options

• For the option \(\overline{AB}\cong\overline{BC}\): An angle bisector of an angle does not imply that the sides adjacent to the bisector (in this case, \(\overline{AB}\) and \(\overline{BC}\)) are congruent. There is no geometric theorem that relates an angle bisector to the congruence of non - angle - side segments in this way.
• For the option “\(B\) is the midpoint of \(\overline{DC}\)”: An angle bisector of \(\angle DBC\) does not mean that \(B\) divides the line segment \(\overline{DC}\) into two equal parts. The angle bisector is about the measure of angles, not the length of line segments on the line containing the side of the angle.
• For the option \(m\angle DBC = 90^{\circ}\): Just because \(\overrightarrow{BA}\) bisects \(\angle DBC\), we have no information about the measure of \(\angle DBC\). It could be any non - negative angle measure (e.g., \(30^{\circ}\), \(60^{\circ}\), \(120^{\circ}\) etc.) that is being bisected.

Answer:

\(m\angle ABD = m\angle ABC\)