Question

the converse of the parallelogram side theorem
given: \\( \overline { l m } \cong \overline { o n } \\) and \\( \overline { l o } \cong \overline { m n } \\)
prove: lmno is a parallelogram.

Explanation:

Step1: Prove triangles congruent

Since \(\overline{LM}\cong\overline{ON}\), \(\overline{LO}\cong\overline{MN}\), and \(\overline{LN}\cong\overline{LN}\) (reflexive property), by the SSS (Side - Side - Side) congruence criterion, \(\triangle LON\cong\triangle NML\).

Step2: Use congruent triangles to find angle congruence

Because \(\triangle LON\cong\triangle NML\), \(\angle LNO\cong\angle NLM\) and \(\angle OLN\cong\angle MNL\).

Step3: Use angle congruence to prove parallel sides

If \(\angle LNO\cong\angle NLM\), then \(LO\parallel MN\) (alternate interior angles are congruent, so lines are parallel). If \(\angle OLN\cong\angle MNL\), then \(LM\parallel ON\) (alternate interior angles are congruent, so lines are parallel).

Step4: Use definition of parallelogram

A quadrilateral with both pairs of opposite sides parallel (\(LO\parallel MN\) and \(LM\parallel ON\)) is a parallelogram.

Answer:

LMNO is a parallelogram.