Question

proving a quadrilateral is a parallelogram
determining necessary information
what additional information would be sufficient, along with the given, to conclude that lmno is a parallelogram? choose two correct answers.
( overline { lo } cong overline { mn } )
( overline { ml } perp overline { lo } )
( overline { ml } parallel overline { no } )

Explanation:

Step1: Recall parallelogram properties

A quadrilateral is a parallelogram if both pairs of opposite sides are parallel (by definition) or if one pair of opposite sides is both parallel and congruent.

Step2: Analyze each option

• For $\overline{LO}\cong\overline{MN}$: If we assume (from the figure, likely some prior - given parallelism, say $\overline{LM}\parallel\overline{ON}$) and then have $\overline{LO}\cong\overline{MN}$, by the theorem that if one pair of opposite sides is parallel and congruent, the quadrilateral is a parallelogram.
• For $\overline{ML}\parallel\overline{NO}$: If we assume (from the figure, likely some prior - given congruent sides, say $\overline{LM}\cong\overline{ON}$) and then have $\overline{ML}\parallel\overline{NO}$, by the theorem that if one pair of opposite sides is parallel and congruent, the quadrilateral is a parallelogram.
• For $\overline{ML}\perp\overline{LO}$: Perpendicularity only gives information about the angle between two sides and does not help in proving the quadrilateral is a parallelogram using the standard parallelogram - proving theorems (like opposite sides parallel/congruent, opposite angles equal, diagonals bisecting each other).

Answer:

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