Question

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

Explanation:

Step1: Recall the definition of an angle bisector

An angle bisector divides an angle into two equal - measure angles.

Step2: Analyze the given situation

Since \(\overrightarrow{BA}\) bisects \(\angle DBC\), by the definition of an angle bisector, \(m\angle ABD=m\angle ABC\).

• For the statement \(m\angle DBC = 90^{\circ}\), there is no information in the problem (just that \(BA\) bisects \(\angle DBC\)) to suggest that \(\angle DBC\) is a right - angle.
• For the statement \(B\) is the mid - point of \(\overline{DC}\), the fact that \(BA\) bisects \(\angle DBC\) has no relation to \(B\) being the mid - point of a line segment (angle bisector is about angles, mid - point is about line segments).
• For the statement \(\overline{AB}\cong\overline{BC}\), angle - bisector property does not imply that the sides forming the angles (in this case, the non - common sides of the angles formed by the bisector) are congruent.

Answer:

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