Question

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

assemble the proof by dragging tiles to
the statements and reasons columns.

Explanation:

Step1: Consider triangles \( \triangle LON \) and \( \triangle NML \)

Given \( \overline{LM}\cong\overline{ON} \), \( \overline{LO}\cong\overline{MN} \), and \( \overline{LN}\cong\overline{NL} \) (common side).

Step2: Prove \( \triangle LON\cong\triangle NML \)

By SSS (Side - Side - Side) congruence criterion.

Step3: Find corresponding angles

\( \angle LON\cong\angle NML \) and \( \angle OLN\cong\angle MNL \) (corresponding parts of congruent triangles).

Step4: Show parallel sides

Since \( \angle LON\cong\angle NML \), \( \overline{LO}\parallel\overline{MN} \) (alternate interior angles). Since \( \angle OLN\cong\angle MNL \), \( \overline{LM}\parallel\overline{ON} \) (alternate interior angles).

Step5: Conclude parallelogram

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

Answer:

Since \( \overline{LM}\cong\overline{ON} \), \( \overline{LO}\cong\overline{MN} \), \( \overline{LN}\cong\overline{NL} \) (SSS for \( \triangle LON\) and \( \triangle NML\)), corresponding angles are congruent leading to \( \overline{LO}\parallel\overline{MN} \) and \( \overline{LM}\parallel\overline{ON} \), so \( LMNO \) is a parallelogram.