Question

proof 3
given: ( abcong bc ) and ( d ) is the midpoint of ( ac )
prove: ( \triangle abdcong\triangle cbd )

Explanation:

Step1: Identify given information

Given \(AB = BC\) (side of the triangles).

Step2: Use mid - point property

Since \(D\) is the mid - point of \(AC\), by the definition of mid - point \(AD = CD\) (another pair of sides of the triangles).

Step3: Consider common side

\(BD\) is a common side for \(\triangle ABD\) and \(\triangle CBD\), so \(BD=BD\).

Step4: Apply SSS congruence

By the SSS (Side - Side - Side) congruence criterion, if three sides of one triangle are equal to three sides of another triangle, then the two triangles are congruent. Here, in \(\triangle ABD\) and \(\triangle CBD\), \(AB = BC\), \(AD = CD\) and \(BD=BD\), so \(\triangle ABD\cong\triangle CBD\).

Answer:

Statement	Reason
\(D\) is the mid - point of \(AC\), so \(AD=CD\)	Definition of mid - point
\(BD = BD\)	Common side
\(\triangle ABD\cong\triangle CBD\)	SSS (Side - Side - Side) congruence criterion