Question

proving a quadrilateral is a parallelogram
given: quadrilateral mnol with \\( \overline { m n } \cong \overline { l o } \\) and \\( \overline { m l } \cong \overline { n o } \\)
prove: mnol is a parallelogram.
complete the paragraph proof.
we are given that \\( \overline { m n } \cong \overline { l o } \\) and \\( \overline { m l } \cong \overline { n o } \\). we can draw in \\( \overline { m o } \\)
because between any two points is a line. by the reflexive property,
\\( \overline { m o } \cong \overline { m o } \\). by sss, \\( \triangle m l o \cong \triangle \\)
\\( \overline { m o } \cong \overline { m o } \\). by sss, \\( \triangle m l o \cong \triangle \\)
and \\( \angle n m o \cong \angle l o m \\) of angles are also
definition. based on the
converse of the alternate interior rem, \\( \overline { m n } \parallel \overline { l o } \\) and
\\( \overline { l m } \parallel \overline { n o } \\). based on the definition of a parallelogram, mnol is a
parallelogram.

Explanation:

Step1: Recall SSS Congruence

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) gives \( \angle LMO\cong\angle NOM\) (these are alternate interior angles for lines \( \overline{LM}\) and \( \overline{NO}\) with transversal \( \overline{MO}\)) and \( \angle NMO\cong\angle LOM\) (these are alternate interior angles for lines \( \overline{MN}\) and \( \overline{LO}\) with transversal \( \overline{MO}\)).

Answer:

The first blank is \( \triangle NOM\), the second blank is \( \angle NOM\), the third blank is alternate interior angles.