Question

given that ba bisects ∠dbc, which statement must be true?
○ m∠abd = m∠abc
○ ab ≅ bc
○ b is the midpoint of dc
○ m∠dbc = 90°

Explanation:

Step1: Recall the definition of an angle - bisector

An angle - bisector is a ray that divides an angle into two equal - measure angles.

Step2: Identify the angle and its bisector

Here, \(\overrightarrow{BA}\) is the bisector of \(\angle DBC\). By the definition of an angle bisector, \(\angle ABD\) and \(\angle ABC\) are the two angles formed by the bisector \(\overrightarrow{BA}\) of \(\angle DBC\). So, \(m\angle ABD=m\angle ABC\).

For option B, \(AB\) is a ray and \(BC\) is a line segment. They are not of the same type of geometric object (a ray has one endpoint and extends infinitely in one direction, a line segment has two endpoints), so \(AB\cong BC\) (which would imply they are congruent line segments) is incorrect.

For option C, just because \(\overrightarrow{BA}\) bisects \(\angle DBC\) does not mean \(B\) is the mid - point of \(\overline{DC}\). The mid - point of a line segment divides the line segment into two equal - length line segments, and angle - bisector property is about angles, not line segments.

For option D, there is no information given to suggest that \(\angle DBC = 90^{\circ}\). The angle - bisector definition only relates to the equality of the two sub - angles formed by the bisector, not the measure of the original angle.

Answer:

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