Question

prove: \\( \triangle abe \cong \triangle cde \\).
note: quadrilateral properties are not permitted in this proof.
step
statement
reason
1
\\( \overline { a b } \parallel \overline { c d } \\)
\\( \overline { b d } \\) bisects \\( \overline { a c } \\)
given
try
type of statement

Explanation:

Step1: Identify alternate interior angles

Since \(\overline{AB}\parallel\overline{CD}\), \(\angle BAE=\angle DCE\) (alternate - interior angles) and \(\angle ABE = \angle CDE\) (alternate - interior angles).

Step2: Use the definition of a bisector

Because \(\overline{BD}\) bisects \(\overline{AC}\), \(AE = CE\) (definition of a bisector: if a line bisects a segment, it divides the segment into two equal parts).

Step3: Apply the AAS (Angle - Angle - Side) congruence criterion

In \(\triangle ABE\) and \(\triangle CDE\), we have \(\angle BAE=\angle DCE\), \(\angle ABE=\angle CDE\), and \(AE = CE\). By the AAS (Angle - Angle - Side) congruence criterion, \(\triangle ABE\cong\triangle CDE\).

Answer:

\(\triangle ABE\cong\triangle CDE\) by the AAS (Angle - Angle - Side) congruence criterion.