Question

given: (overline{ab}congoverline{cb}), (overline{bd}) bisects (angle abc) prove: (\triangle abdcong\triangle cbd) 1. (overline{ab}congoverline{cb}) 1. given 2. (overline{bd}) bisects (angle abc) 2. given 3. (angle abdcongangle cbd) 4. (overline{ad}congoverline{bd}) 5. (\triangle abdcong\triangle cbd)

Explanation:

Step1: Definition of Angle Bisector

Since \(BD\) bisects \(\angle ABC\), by the definition of an angle bisector, \(\angle ABD=\angle CBD\).

Step2: Common Side

\(\overline{BD}\) is a common side to both \(\triangle ABD\) and \(\triangle CBD\), so \(BD = BD\) (reflexive property of equality).

Step3: SAS Congruence Criterion

We have \(\overline{AB}\cong\overline{CB}\) (given), \(\angle ABD\cong\angle CBD\) (from step 1), and \(\overline{BD}\cong\overline{BD}\) (from step 2). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle ABD\cong\triangle CBD\).

Answer:

\(\triangle ABD\cong\triangle CBD\) by the SAS (Side - Angle - Side) congruence criterion.