Question

in quadrilateral lmno, (overline{lo}paralleloverline{mn}). what additional information would be sufficient, along with the given, to conclude that lmno is a parallelogram? check all that apply. (squareoverline{ml}paralleloverline{no}) (squareoverline{ml}perpoverline{lo}) (squareoverline{lo}congoverline{mn}) (squareoverline{ml}congoverline{lo}) (squareoverline{mn}perpoverline{no})

Explanation:

Step1: Recall parallelogram properties

A quadrilateral is a parallelogram if one - pair of opposite sides is parallel and congruent or if both pairs of opposite sides are parallel.

Step2: Analyze each option

• Option 1: If $\overline{ML}\parallel\overline{NO}$ and $\overline{LO}\parallel\overline{MN}$, then by the definition of a parallelogram (a quadrilateral with both pairs of opposite sides parallel), LMNO is a parallelogram.
• Option 2: $\overline{ML}\perp\overline{LO}$ only gives information about the angle between two adjacent sides and does not help in proving it is a parallelogram.
• Option 3: If $\overline{LO}\cong\overline{MN}$ and $\overline{LO}\parallel\overline{MN}$, then one - pair of opposite sides is parallel and congruent, so LMNO is a parallelogram.
• Option 4: $\overline{ML}\cong\overline{LO}$ gives information about two adjacent sides and is not sufficient to prove it is a parallelogram.
• Option 5: $\overline{MN}\perp\overline{NO}$ only gives information about the angle between two adjacent sides and does not help in proving it is a parallelogram.

Answer:

$\overline{ML}\parallel\overline{NO}$, $\overline{LO}\cong\overline{MN}$