Question

1. in the figure, \\( \triangle b a t \cong \triangle c a t \\). which statement is not true by cpctc?

\\( \angle c a b \cong \angle s a b \\)
\\( \angle c a s \cong \angle b a s \\)
\\( \overline{t a} \cong \overline{a t} \\)
\\( \overline{t b} \cong \overline{t c} \\)

Explanation:

Step1: Recall CPCTC

CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent". If \(\triangle BAT\cong\triangle CAT\), then their corresponding sides and angles are congruent.

Step2: Analyze each option

• For \(\angle CAS=\angle BAS\): Since \(\triangle BAT\cong\triangle CAT\), \(\angle BAT=\angle CAT\). If we consider the angle - addition postulate and the fact that \(\angle BAS\) and \(\angle CAS\) are parts of \(\angle BAT\) and \(\angle CAT\) respectively (by the common side \(AS\)), \(\angle CAS=\angle BAS\) (by CPCTC).
• For \(\overline{TA}\cong\overline{AT}\): This is the reflexive property of congruence (\(TA = AT\) as it is the same segment, but in the context of congruent triangles \(\triangle BAT\) and \(\triangle CAT\), \(TA\) in \(\triangle BAT\) corresponds to \(AT\) in \(\triangle CAT\) (they are the same length, just written in reverse order).
• For \(\overline{TB}\cong\overline{TC}\): In \(\triangle BAT\) and \(\triangle CAT\), \(TB\) and \(TC\) are corresponding sides. So, \(\overline{TB}\cong\overline{TC}\) (by CPCTC).
• For \(\angle CAB=\angle SAB\): There is no information from \(\triangle BAT\cong\triangle CAT\) that would make \(\angle CAB\) and \(\angle SAB\) congruent. \(\angle CAB\) is not a corresponding part of \(\triangle BAT\) and \(\triangle CAT\) in a way that would imply this congruence by CPCTC.

Answer:

\(\angle CAB\cong\angle SAB\)