Question

given that \\( \overrightarrow{ba} \\) bisects \\( \angle dbc \\), which statement must be true? \\( m \angle abd=m \angle abc \\) \\( \overline{ab} \cong \overline{bc} \\) b is the midpoint of \\( \overline{dc} \\). \\( 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. Since \(\overrightarrow{BA}\) bisects \(\angle DBC\), it means that \(\angle ABD\) 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 the other options

• For the option \(\overline{AB}\cong\overline{BC}\): Just because \(\overrightarrow{BA}\) bisects \(\angle DBC\), there is no information given to suggest that the lengths of \(AB\) and \(BC\) are equal. An angle bisector does not imply that the sides adjacent to the angle (in this non - triangle context related to the angle bisector property) are congruent.
• 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 \(DC\) into two equal - length parts. The angle bisector property is about angles, not about the mid - point of a line segment.
• For the option \(m\angle DBC = 90^{\circ}\): There is no indication in the problem statement or the diagram (other than the perpendicular - looking line \(BA\) which is not relevant to the measure of \(\angle DBC\) in terms of being a right angle based on the angle - bisector definition) that \(\angle DBC\) is a right angle. The angle - bisector definition only relates to the equality of the two sub - angles formed by the bisector, not to the measure of the original angle.

Answer:

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