Question

given: quadrilateral mnol with $overline{mn}congoverline{lo}$ and $overline{ml}congoverline{no}$ prove: mnol is a parallelogram. complete the paragraph proof. we are given that $overline{mn}congoverline{lo}$ and $overline{ml}congoverline{no}$. we can draw in $overline{mo}$ because between any two points is a line. by the reflexive property, $overline{mo}congoverline{mo}$. by sss, $\triangle{mlo}cong\triangle{\text{ }}$. by cpctc, $angle{lmo}congangle{\text{ }}$ and $angle{nmo}congangle{lom}$. both pairs of angles are also $\text{ }$ based on the definition. alternate interior angles alternate exterior angles corresponding angles. mnol is a parallelogram.

Explanation:

Step1: Identify congruent triangles

Since $\overline{MN}\cong\overline{LO}$, $\overline{ML}\cong\overline{NO}$ and $\overline{MO}\cong\overline{MO}$ (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) gives $\angle LMO\cong\angle NOM$ and $\angle NMO\cong\angle LOM$.

Step3: Determine angle type

These pairs of angles are alternate interior angles. When alternate interior angles are congruent, the opposite sides are parallel. Since $\angle LMO\cong\angle NOM$, $ML\parallel NO$ and since $\angle NMO\cong\angle LOM$, $MN\parallel LO$. A quadrilateral with both pairs of opposite sides parallel is a parallelogram.

Answer:

1. $\triangle NOM$
2. $\angle NOM$
3. alternate interior angles