Question

proving the converse of the parallelogram side theorem
given: lm ≅ on and lo ≅ mn
prove: lmno is a parallelogram.
assemble the proof by dragging tiles to the statements and reasons columns

Explanation:

Step1: Draw diagonal LN

This creates two triangles $\triangle LMN$ and $\triangle ONL$.

Step2: Prove $\triangle LMN\cong\triangle ONL$

We know that $LM\cong ON$ and $LO\cong MN$ (given). Also, $LN = LN$ (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 - side relationships

$\angle MLN\cong\angle ONL$ implies $LM\parallel ON$ (alternate interior angles are equal, so lines are parallel). Similarly, $\angle LMN\cong\angle NOL$ implies $LO\parallel MN$.

Step5: Define parallelogram

A quadrilateral with both pairs of opposite sides parallel is a parallelogram. So, $LMNO$ is a parallelogram.

Answer:

LMNO is a parallelogram.