Question

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

Explanation:

Step1: Alternate interior angles

Since \( \overline{AB}\parallel\overline{CD}\), then \( \angle BAC\cong\angle DCA\) (alternate - interior angles) and \( \angle ABD\cong\angle CDB\) (alternate - interior angles).

Step2: Opposite sides of parallelogram

In parallelogram \(ABCD\), \( \overline{AB}\cong\overline{CD}\) (opposite sides of a parallelogram are congruent).

Step3: ASA congruence

In \(\triangle ABE\) and \(\triangle CDE\), we have \(\angle BAC\cong\angle DCA\), \(\overline{AB}\cong\overline{CD}\), \(\angle ABD\cong\angle CDB\). So, \(\triangle ABE\cong\triangle CDE\) (by ASA - Angle - Side - Angle congruence criterion).

Step4: Corresponding parts of congruent triangles

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

Answer:

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