Question

1. in the figure, $\triangle abd\cong\triangle cbd$ by angle - side - angle (asa). which segments are congruent by cpctc?

$\overline{ab}\cong\overline{cd}$
$\overline{bc}\cong\overline{ad}$
$\overline{db}\cong\overline{dc}$

Explanation:

Step1: Recall CPCTC

CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent".

Step2: Identify corresponding sides

Since \(\triangle ABD\cong\triangle CBD\) by ASA, their corresponding sides are congruent.

• In \(\triangle ABD\) and \(\triangle CBD\), \(AB\) corresponds to \(CB\) and \(AD\) corresponds to \(CD\), and \(BD\) is common.
• Among the given options, \(\overline{AB}\) and \(\overline{CD}\) are not corresponding sides. \(\overline{BC}\) and \(\overline{AD}\) are not corresponding sides.
• But if we consider the congruence \(\triangle ABD\cong\triangle CBD\), we know that \(AB = CB\), \(AD=CD\) and \(BD = BD\). Wait, no, actually, if \(\triangle ABD\cong\triangle CBD\), then \(AB\) corresponds to \(CB\), \(BD\) is common and \(AD\) corresponds to \(CD\). Wait, no, let's use the order of congruence. When \(\triangle ABD\cong\triangle CBD\), the vertices \(A\) corresponds to \(C\), \(B\) corresponds to \(B\) (common vertex) and \(D\) corresponds to \(D\) (common vertex). So \(AB\) (side between \(A\) and \(B\)) corresponds to \(CB\) (side between \(C\) and \(B\)), \(BD\) (side between \(B\) and \(D\)) is common, \(AD\) (side between \(A\) and \(D\)) corresponds to \(CD\) (side between \(C\) and \(D\)).

Wait, no, let's re - check. The ASA congruence: assume \(\angle ABD=\angle CBD\), \(BD = BD\) (common side) and \(\angle ADB=\angle CDB\). Then by ASA \(\triangle ABD\cong\triangle CBD\). So the sides: \(AB\) (opposite to \(\angle ADB\)) corresponds to \(CB\) (opposite to \(\angle CDB\)), \(AD\) (opposite to \(\angle ABD\)) corresponds to \(CD\) (opposite to \(\angle CBD\)) and \(BD\) is common.

Answer:

None of the given options are correct. But if we assume a typo in the problem (maybe \(\triangle ABD\cong\triangle CDB\) instead of \(\triangle CBD\)), then \(\overline{AB}\cong\overline{CD}\) and \(\overline{AD}\cong\overline{CB}\). But based on the given \(\triangle ABD\cong\triangle CBD\) (vertex order), there is an error in the options. However, if we consider the most probable intended answer (assuming a mis - labeling of the triangle congruence in the problem statement), if we assume the congruence is \(\triangle ABD\cong\triangle CDB\) (so \(A\) to \(C\), \(B\) to \(D\), \(D\) to \(B\)), then \(\overline{AB}\cong\overline{CD}\)

So, if we go with the problem's likely intended congruence (despite vertex - order typo), the answer is \(\overline{AB}\cong\overline{CD}\)