Question

question
fill in the missing statement and reason of the proof below.
given: \\( \overline { a b } \cong \overline { b c } \\) and \\( \angle a b d \cong \angle c b d \\).
prove: \\( \overline { b d } \\) bisects \\( \overline { a c } \\) and \\( \overline { b d } \perp \overline { a c } \\).

Explanation:

Step1: Analyze the goal of the proof

We need to prove that \( \overline{BD} \) bisects \( \overline{AC} \). By the definition of a bisector of a segment, if a point \( D \) lies on \( \overline{AC} \) and \( \overline{AD}\cong\overline{DC} \), then \( D \) is the mid - point of \( \overline{AC} \), and \( \overline{BD} \) bisects \( \overline{AC} \).

Answer:

Statement: \( \overline{BD} \) bisects \( \overline{AC} \); Reason: If a segment divides another segment into two congruent segments, then it bisects the segment.