Question

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

Explanation:

Step1: Recall Parallelogram Theorem

A quadrilateral is a parallelogram if one pair of opposite sides are both parallel and congruent. We know \(\overline{AB} \parallel \overline{CD}\) from the problem.

Step2: Analyze Each Option

• Option 1: \(\overline{AB} \cong \overline{CD}\). Since we already have \(\overline{AB} \parallel \overline{CD}\), by the theorem (one pair of opposite sides parallel and congruent implies parallelogram), this would make \(ABDC\) a parallelogram.
• Option 2: \(\overline{AC} \cong \overline{BD}\) (the diagonals being congruent) is a property of rectangles, not a general parallelogram, and doesn't help here as we need info about the sides we know are parallel.
• Option 3: \(\overline{AB} \perp \overline{BD}\) (a side perpendicular to another) doesn't relate to the parallelogram definition for the given parallel sides.
• Option 4: \(\overline{CD} \perp \overline{BD}\) (a side perpendicular to another) also doesn't relate to the parallelogram definition for the given parallel sides.

Answer:

\(\boldsymbol{\overline{AB} \cong \overline{CD}}\) (the first option: \(\overline{AB} \cong \overline{CD}\))