Question

1. in the figure, \\( \triangle bat \cong \triangle cat \\). which statement is not true by cpctc?

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

Explanation:

Step1: Recall CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

If \(\triangle BAT\cong\triangle CAT\), then their corresponding sides and angles are congruent.

• For sides: \(AB = AC\) (\(\overline{AB}\cong\overline{AC}\)), \(BT = CT\), \(AT=AT\) (common side)
• For angles: \(\angle BAT=\angle CAT\), \(\angle BTA=\angle CTA\), \(\angle ABT=\angle ACT\)

Step2: Analyze each option

• Option \(\overline{AB}\cong\overline{AC}\): Since \(\triangle BAT\cong\triangle CAT\), \(AB\) and \(AC\) are corresponding sides. So, \(\overline{AB}\cong\overline{AC}\) is True by CPCTC.
• Option \(\angle BAT=\angle CAT\): \(\angle BAT\) and \(\angle CAT\) are corresponding angles of \(\triangle BAT\) and \(\triangle CAT\). So, \(\angle BAT = \angle CAT\) is True by CPCTC.
• Option \(\angle BTA=\angle CTA\): \(\angle BTA\) and \(\angle CTA\) are corresponding angles of \(\triangle BAT\) and \(\triangle CAT\). So, \(\angle BTA=\angle CTA\) is True by CPCTC.
• Option \(\overline{CS}\cong\overline{ST}\): There is no information from \(\triangle BAT\cong\triangle CAT\) that directly gives a relationship between \(CS\) and \(ST\). \(S\) is a point on \(BC\) (assuming from the figure, but not part of the congruent - triangle correspondence for \(\triangle BAT\) and \(\triangle CAT\))

Answer:

\(\overline{CS}\cong\overline{ST}\)