Question

in quadrilateral abdc, \\( \overline{ab} \parallel \overline{cd} \\). which additional piece of information is needed to determine that abdc is a parallelogram? \\( \overline{ab} \cong \overline{cd} \\) \\( \overline{ac} \cong \overline{bd} \\) \\( \overline{ab} \perp \overline{bd} \\) \\( \overline{cd} \perp \overline{bd} \\)

Explanation:

Step1: Recall the properties of a parallelogram

One of the properties of a parallelogram is that if one pair of opposite sides is both parallel and congruent, then the quadrilateral is a parallelogram.
We are given that \(\overline{AB}\parallel\overline{CD}\). If we also have \(\overline{AB}\cong\overline{CD}\), then by the parallelogram property (a quadrilateral with one pair of opposite sides parallel and congruent is a parallelogram), quadrilateral \(ABDC\) is a parallelogram.

Step2: Analyze other options

• For \(\overline{AC}\cong\overline{BD}\): Congruent diagonals (\(\overline{AC}\) and \(\overline{BD}\)) do not guarantee that a quadrilateral with \(\overline{AB}\parallel\overline{CD}\) is a parallelogram. Isosceles trapezoids (a non - parallelogram) can have congruent diagonals.
• For \(\overline{AB}\perp\overline{BD}\): A right angle between \(\overline{AB}\) and \(\overline{BD}\) along with \(\overline{AB}\parallel\overline{CD}\) does not give us information about the relationship of the other pair of sides (\(\overline{AC}\) and \(\overline{BD}\)) in terms of parallelism or the relationship of \(\overline{AB}\) and \(\overline{CD}\) in terms of length.
• For \(\overline{CD}\perp\overline{BD}\): A right angle between \(\overline{CD}\) and \(\overline{BD}\) along with \(\overline{AB}\parallel\overline{CD}\) does not give us information about the relationship of the other pair of sides (\(\overline{AC}\) and \(\overline{BD}\)) in terms of parallelism or the relationship of \(\overline{AB}\) and \(\overline{CD}\) in terms of length.

Answer:

\(\overline{AB}\cong\overline{CD}\)