Question

prove: \\( \triangle abe \cong \triangle cde \\).
note: quadrilateral properties are not permitted in this proof.

step
statement
reason
1
\\( \overline { b d } \\) and \\( \overline { a c } \\) bisect each other
given
2
\\( \angle a e b \cong \angle c e d \\)
vertical angles are congruent

Explanation:

Step1: Define the property of bisecting

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

Step2: Apply the SAS (Side - Angle - Side) congruence criterion

In \(\triangle ABE\) and \(\triangle CDE\), we have:

• \(AE = CE\) (from Step 1)
• \(\angle AEB\cong\angle CED\) (Given as vertical angles are congruent)
• \(BE = DE\) (from Step 1)

By the SAS congruence criterion (\(SAS\): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent), \(\triangle ABE\cong\triangle CDE\).

Answer:

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