Question

using the converse of the parallelogram side theorem
given: $overline{lm}congoverline{on}$ and $overline{lo}congoverline{mn}$
prove: lmno is a parallelogram.
assemble the proof by dragging tiles to the statements and reasons columns.

Explanation:

Step1: Draw diagonal LN

Draw a line segment LN to divide the quadrilateral LMNO into two triangles, $\triangle LMN$ and $\triangle ONL$.

Step2: Prove triangle - congruence

In $\triangle LMN$ and $\triangle ONL$, we have $\overline{LM}\cong\overline{ON}$, $\overline{LO}\cong\overline{MN}$ and $\overline{LN}\cong\overline{LN}$ (common side). By SSS (Side - Side - Side) congruence criterion, $\triangle LMN\cong\triangle ONL$.

Step3: Find corresponding angles

Since $\triangle LMN\cong\triangle ONL$, $\angle MLN\cong\angle ONL$ and $\angle LMN\cong\angle NOL$ (corresponding parts of congruent triangles are congruent).

Step4: Use angle - congruence for parallel sides

$\angle MLN\cong\angle ONL$ implies that $\overline{LM}\parallel\overline{ON}$ (alternate interior angles are congruent, then the lines are parallel). Also, $\angle LMN\cong\angle NOL$ implies that $\overline{LO}\parallel\overline{MN}$.

Step5: Apply parallelogram definition

Since both pairs of opposite sides of quadrilateral LMNO are parallel, LMNO is a parallelogram (a quadrilateral is a parallelogram if both pairs of opposite sides are parallel).

Answer:

LMNO is a parallelogram.