Question

1. in the figure, \\( \triangle bat \cong \triangle cat \\). which statement is true by cpctc?

\\( \overline { bt } \cong \overline { ct } \\)
\\( \overline { sb } \cong \overline { bc } \\)
\\( \angle abs = \angle sct \\)
\\( \angle bat = \angle cta \\)

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 \(\overline{BT}\cong\overline{CT}\): In \(\triangle BAT\) and \(\triangle CAT\), \(\overline{BT}\) and \(\overline{CT}\) are corresponding sides. Since \(\triangle BAT\cong\triangle CAT\), by CPCTC, \(\overline{BT}\cong\overline{CT}\).
• For \(\overline{SB}\cong\overline{BC}\): There is no information from \(\triangle BAT\cong\triangle CAT\) that directly gives this congruence. \(S\) is a point of intersection, and this is not a corresponding part of the two congruent triangles \(\triangle BAT\) and \(\triangle CAT\).
• For \(\angle ABS\cong\angle SCT\): These angles are not corresponding angles of \(\triangle BAT\) and \(\triangle CAT\).
• For \(\angle BAT\cong\angle CTA\): \(\angle BAT\) is an angle of \(\triangle BAT\) and \(\angle CTA\) is an angle of \(\triangle CAT\), but they are not corresponding angles. Corresponding angles of \(\triangle BAT\) and \(\triangle CAT\) would be \(\angle BAT\) and \(\angle CAT\), \(\angle ABT\) and \(\angle ACT\), \(\angle ATB\) and \(\angle ATC\)

Answer:

\(\overline{BT}\cong\overline{CT}\)