Question

given: (overline{eb}) bisects (angle abc) and (angle ccongangle a) prove: (\triangle cbdcong\triangle abd)

Explanation:

Step1: Use the definition of angle - bisector

Since \(BD\) bisects \(\angle ABC\), we have \(\angle ABD=\angle CBD\) (Definition of angle - bisector).

Step2: Identify the given angle equality

Given \(\angle C = \angle A\) (Given).

Step3: Use the common side

\(BD = BD\) (Reflexive property of equality: a segment is congruent to itself).

Step4: Apply the AAS (Angle - Angle - Side) congruence criterion

In \(\triangle CBD\) and \(\triangle ABD\), we have \(\angle C=\angle A\), \(\angle CBD = \angle ABD\), and \(BD=BD\). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle CBD\cong\triangle ABD\).

Answer:

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