Question

which of the following statements would we need to prove \\( \triangle a b c \cong \triangle d e f \\) by asa?
\\( \overline{b c} \cong \overline{e f} \\)
\\( \overline{a c} \cong \overline{d f} \\)
\\( \angle a \cong \angle d \\)
\\( \overline{a b} \cong \overline{d e} \\)

Explanation:

Step1: Recall ASA (Angle - Side - Angle) criterion

ASA states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.
In \(\triangle ABC\) and \(\triangle DEF\), we already have \(\angle C\cong\angle F\) and \(\angle B\cong\angle E\). The included side between \(\angle B\) and \(\angle C\) in \(\triangle ABC\) is \(BC\), and the included side between \(\angle E\) and \(\angle F\) in \(\triangle DEF\) is \(EF\).

Step2: Check the congruence of the included sides

If \(\overline{BC}\cong\overline{EF}\), then by ASA criterion \(\triangle ABC\cong\triangle DEF\)

Answer:

\(\overline{BC}\cong\overline{EF}\)