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 parallelogram properties

A quadrilateral with one pair of parallel sides (\(\overline{AB}\parallel\overline{CD}\)) can be a parallelogram if that pair of parallel sides is also congruent.

Step2: Analyze each option

• For \(\overline{AB}\cong\overline{CD}\): By the theorem, if a quadrilateral has one pair of sides that are both parallel and congruent, then it is a parallelogram.
• For \(\overline{AC}\cong\overline{BD}\): Congruent diagonals (\(\overline{AC}\) and \(\overline{BD}\)) do not guarantee a parallelogram. For example, an isosceles trapezoid (which is not a parallelogram) can have congruent diagonals.
• For \(\overline{AB}\perp\overline{BD}\): A right - angle between \(\overline{AB}\) and \(\overline{BD}\) only gives information about the angle at \(B\) related to \(AB\) and \(BD\), not about the nature of the quadrilateral as a parallelogram.
• For \(\overline{CD}\perp\overline{BD}\): A right - angle between \(\overline{CD}\) and \(\overline{BD}\) only gives information about the angle at \(D\) related to \(CD\) and \(BD\), not about the nature of the quadrilateral as a parallelogram.

Answer:

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