Question

state the third congruence that must be given to prove that \\( \triangle abc \cong \triangle def \\) using sas congruence postulate. \\( \angle b \cong \angle e \\) \\( \overline{bc} \cong \overline{ef} \\)

Explanation:

Step1: Recall SAS Congruence Postulate

The SAS (Side - Angle - Side) Congruence Postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Step2: Identify the given parts

We are given that \(\angle B\cong\angle E\). For the SAS postulate, we need two sides and the included angle. The sides adjacent to \(\angle B\) in \(\triangle ABC\) are \(AB\) and \(BC\), and the sides adjacent to \(\angle E\) in \(\triangle DEF\) are \(DE\) and \(EF\).

Step3: Determine the missing side

Since \(\angle B\) is the included angle between \(AB\) and \(BC\) in \(\triangle ABC\), and \(\angle E\) is the included angle between \(DE\) and \(EF\) in \(\triangle DEF\), and we know \(\angle B\cong\angle E\), if we assume \(AB\cong DE\) (the other pair of sides adjacent to the given congruent angles), then by the SAS postulate \(\triangle ABC\cong\triangle DEF\).

Answer:

\(AB\cong DE\)