Question

1. in the figure, \\( \triangle b a t \cong \triangle c a t \\). which statement is true by cpctc?

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

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 \(\overline{BA}\cong\overline{CA}\): In \(\triangle BAT\) and \(\triangle CAT\), \(BA\) and \(CA\) are corresponding sides. Since \(\triangle BAT\cong\triangle CAT\), by CPCTC, \(\overline{BA}\cong\overline{CA}\).
• For \(\angle ABS\cong\angle SCT\): These angles are not corresponding angles of \(\triangle BAT\) and \(\triangle CAT\).
• For \(\overline{SB}\cong\overline{BC}\): These segments are not corresponding parts of \(\triangle BAT\) and \(\triangle CAT\).
• For \(\angle BAT\cong\angle CTA\): These angles are not corresponding angles of \(\triangle BAT\) and \(\triangle CAT\).

Answer:

\(\overline{BA}\cong\overline{CA}\)