Question

statement reason
1 $overleftrightarrow{ac}perpoverleftrightarrow{bd}$ definition of perpendicular bisector.
2 $overline{ab}congoverline{cb}$ definition of perpendicular bisector.
3 $angle abd& angle cbd$ are both right angles definition of perpendicular.
4 $overline{bd}congoverline{bd}$ line segments are congruent to themselves.
5 $\triangle abdcong\triangle cbd$ congruency postulate (2, 3, 4).
6 corresponding parts of congruent triangles have the same measure.

Explanation:

Step1: Identify congruent - side, angle, side

From statement 2 $\overline{AB}\cong\overline{CB}$, statement 3 ($\angle ABD$ and $\angle CBD$ are right - angles, so $\angle ABD\cong\angle CBD$) and statement 4 $\overline{BD}\cong\overline{BD}$, we can use the Side - Angle - Side (SAS) congruency postulate.

Step2: Determine missing statement

Since $\triangle ABD\cong\triangle CBD$, by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem, we can say that $\angle ADB\cong\angle CDB$ (or other corresponding parts like $\overline{AD}\cong\overline{CD}$). Let's assume we are looking for an angle - related statement, so $\angle ADB\cong\angle CDB$.

Answer:

1. Reason: SAS congruency postulate
2. Statement: $\angle ADB\cong\angle CDB$