Question

given: abcd is a parallelogram.
diagonals \\( \overline { a c } \\), bd intersect at e.
prove: \\( \overline { a e } \cong \overline { c e } \\) and \\( \overline { b e } \cong \overline { d e } \\)

Explanation:

Step1: Identify alternate interior angles

Since \(AB\parallel CD\), \(\angle BAE\cong\angle DCE\) and \(\angle ABE\cong\angle CDE\) (by the definition of alternate - interior angles).

Step2: Prove \(\triangle ABE\cong\triangle CDE\)

We know \(AB\cong CD\) (from step 3 of the given), \(\angle BAE\cong\angle DCE\), \(\angle ABE\cong\angle CDE\). By the Angle - Angle - Side (AAS) congruence criterion, \(\triangle ABE\cong\triangle CDE\).

Step3: Use congruent triangles to find segment congruence

Since \(\triangle ABE\cong\triangle CDE\), corresponding parts of congruent triangles are congruent. So, \(AE\cong CE\) and \(BE\cong DE\).

Answer:

\(\overline{AE}\cong\overline{CE}\) and \(\overline{BE}\cong\overline{DE}\) are proved as above.