Question

given: \\( \overline { a c } \\) bisects \\( \overline { b d } \\) and \\( \overline { b c } \parallel \overline { a d } \\).
prove: \\( \triangle b e c \cong \triangle d e a \\).

note: \\( \overline { a c } \\) and \\( \overline { b d } \\) are segments

Explanation:

Step1: Find equal segments

Since \( \overline{AC} \) bisects \( \overline{BD} \), by the definition of a bisector, \( BE = DE \).

Step2: Find equal angles

Because \( \overline{BC}\parallel\overline{AD} \), by the alternate - interior angles theorem, \( \angle EBC=\angle EDA \) and \( \angle ECB=\angle EAD \).

Step3: Prove congruence

In \( \triangle BEC \) and \( \triangle DEA \), we have \( BE = DE \) (from Step 1), \( \angle EBC=\angle EDA \), \( \angle ECB=\angle EAD \). By the Angle - Angle - Side (AAS) congruence criterion, \( \triangle BEC\cong\triangle DEA \).

Answer:

\( \triangle BEC\cong\triangle DEA \) (by AAS)