Question

in quadrilateral lmno, \\( \overline { lo } \parallel \overline { mn } \\).
what additional information would be sufficient, along
with the given, to conclude that lmno is a
parallelogram? check all that apply.
\\( \square \overline { ml } \parallel \overline { no } \\)
\\( \square \overline { ml } \perp \overline { lo } \\)
\\( \square \overline { lo } \cong \overline { mn } \\)
\\( \square \overline { ml } \cong \overline { lo } \\)
\\( \square \overline { mn } \perp \overline { no } \\)

Explanation:

• For \(\overline{ML}\parallel\overline{NO}\):
• A quadrilateral with both pairs of opposite sides parallel (\(\overline{LO}\parallel\overline{MN}\) and \(\overline{ML}\parallel\overline{NO}\)) is a parallelogram by the definition of a parallelogram.
• For \(\overline{LO}\cong\overline{MN}\):
• A quadrilateral with one pair of opposite sides both parallel (\(\overline{LO}\parallel\overline{MN}\)) and congruent (\(\overline{LO}\cong\overline{MN}\)) is a parallelogram. This is a well - known parallelogram theorem.
• For \(\overline{ML}\perp\overline{LO}\) and \(\overline{MN}\perp\overline{NO}\):
• Just because one pair of adjacent sides is perpendicular does not guarantee that the quadrilateral is a parallelogram. A trapezoid (which is not a parallelogram in general) can have right angles.
• For \(\overline{ML}\cong\overline{LO}\):
• Knowing that two adjacent sides are congruent (\(\overline{ML}\cong\overline{LO}\)) along with \(\overline{LO}\parallel\overline{MN}\) is not sufficient to prove that the quadrilateral is a parallelogram. There is no theorem that states that a quadrilateral with one pair of parallel sides and two adjacent sides congruent is a parallelogram.

Answer:

\(\overline{ML}\parallel\overline{NO}\), \(\overline{LO}\cong\overline{MN}\)