Question

given: \\( \overline { b d } \\) and \\( \overline { a c } \\) bisect each other.
prove: \\( \triangle a b e \cong \triangle c d e \\).
note: quadrilateral properties are not permitted in this proof.

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

Explanation:

Step1: Definition of bisect

Since \(\overline{BD}\) and \(\overline{AC}\) bisect each other, we have \(AE = CE\) and \(BE=DE\)

Step2: Vertical angles

\(\angle AEB\) and \(\angle CED\) are vertical angles. By the vertical angles theorem, \(\angle AEB\cong\angle CED\)

Step3: SAS (Side - Angle - Side) congruence criterion

In \(\triangle ABE\) and \(\triangle CDE\), we have \(AE = CE\), \(\angle AEB\cong\angle CED\), and \(BE = DE\). By the SAS congruence criterion, \(\triangle ABE\cong\triangle CDE\)

Answer:

\(\triangle ABE\cong\triangle CDE\) (by SAS)