Question

given: \\( \overline{ab} \cong \overline{ac} \\) and \\( \triangle bde \cong \triangle cde \\).
prove: \\( \angle bad \cong \angle cad \\).
note: quadrilateral properties are not permitted in this proof.
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$$\begin{tabular}{|c|c|} \\hline step & \\\\ \\hline statement & \\( \\overline{ab} \\cong \\overline{ac} \\) \\\\ \\hline reason & given \\\\ \\hline 1 & \\( \\triangle bde \\cong \\triangle cde \\) \\\\ \\hline \\end{tabular}$$

Explanation:

Step1: Use the property of congruent triangles

Since \(\triangle BDE\cong\triangle CDE\), then \(BD = CD\) (corresponding parts of congruent triangles are equal).

Step2: Prove \(\triangle ABD\cong\triangle ACD\)

We know that \(AB\cong AC\) (given), \(BD = CD\) (from step 1), and \(AD = AD\) (common side). By the SSS (Side - Side - Side) congruence criterion, \(\triangle ABD\cong\triangle ACD\).

Step3: Use the property of congruent triangles again

Since \(\triangle ABD\cong\triangle ACD\), then \(\angle BAD\cong\angle CAD\) (corresponding parts of congruent triangles are equal).

Answer:

\(\angle BAD\cong\angle CAD\) is proved.