Question

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

Explanation:

Step1: Identify congruent triangles

We are given \( \overline{MN}\cong\overline{LO}\), \( \overline{ML}\cong\overline{NO}\) 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). So, \( \angle LMO\cong\angle NOM\).

Step3: Determine angle type

\( \angle LMO\) and \( \angle NOM\) (also \( \angle NMO\) and \( \angle LOM\)) are alternate interior angles.

Answer:

\(\triangle NOM\), \(\angle NOM\), alternate interior angles