Question

given: quadrilateral mnol with mn ≅ lo and ml ≅ no. prove: mnol is a parallelogram. complete the paragraph proof. we are given that mn ≅ lo and ml ≅ no. we can draw in mo because between any two points is a line. by the reflexive property, mo ≅ mo. by sss, △mlo ≅ △__. by cpctc, ∠lmo ≅ ∠ and ∠nmo ≅ ∠lom. both pairs of angles are also __ based on the definition. based on the converse of the alternate interior angles theorem, mn ∥ lo and lm ∥ no. based on the definition of a parallelogram, mnol is a parallelogram.

Explanation:

Step1: Identify congruent triangles

Since $MN\cong LO$, $ML\cong NO$ and $MO\cong MO$ (reflexive property), by SSS (Side - Side - Side) congruence criterion, $\triangle MLO\cong\triangle NOM$.

Step2: Use CPCTC

CPCTC (Corresponding Parts of Congruent Triangles are Congruent). So $\angle LMO\cong\angle NOM$ and $\angle NMO\cong\angle LOM$.

Step3: Determine angle relationship

These pairs of angles are alternate interior angles.

Step4: Apply converse of alternate - interior angles theorem

Since $\angle LMO\cong\angle NOM$ and $\angle NMO\cong\angle LOM$ (alternate interior angles), by the converse of the alternate interior angles theorem, $MN\parallel LO$ and $LM\parallel NO$.

Step5: Use parallelogram definition

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

Answer:

$\triangle MLO\cong\triangle NOM$; $\angle NOM$; alternate interior angles; $MNOL$ is a parallelogram.