in quadrilateral abdc, $\overline{ab} \parallel \overline{cd}$. which additional piece of information is needed to determine that abdc is a parallelogram?

$\bigcirc$ $\overline{ab} \cong \overline{cd}$
$\bigcirc$ $\overline{ac} \cong \overline{bd}$
$\bigcirc$ $\overline{ab} \perp \overline{bd}$
$\bigcirc$ $\overline{cd} \perp \overline{bd}$

Question

in quadrilateral abdc, $\overline{ab} \parallel \overline{cd}$. which additional piece of information is needed to determine that abdc is a parallelogram?

$\bigcirc$ $\overline{ab} \cong \overline{cd}$
$\bigcirc$ $\overline{ac} \cong \overline{bd}$
$\bigcirc$ $\overline{ab} \perp \overline{bd}$
$\bigcirc$ $\overline{cd} \perp \overline{bd}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Parallelogram Proof Theorems", "Quadrilateral Classification" ], "current_concepts": [ "Parallelogram Properties", "Parallelogram Proof Theorems", "Quadrilateral Classification" ] } ## Analyze the given geometric properties $$ \begin{aligned} &\text{Given: Quadrilateral } ABDC \text{ with } \overline{AB} \parallel \overline{CD}.\ &\text{Goal: Prove } ABDC \text{ is a parallelogram.} \end{aligned} $$ ## Evaluate the sufficient conditions for a parallelogram $$ \begin{aligned} &\text{A quadrilateral is a parallelogram if:} \ &1. \text{ Both pairs of opposite sides are parallel (definition).} \ &2. \text{ One pair of opposite sides is both parallel and congruent.} \end{aligned} $$ ## Match with the given options $$ \begin{aligned} &\text{Since } \overline{AB} \parallel \overline{CD} \text{ is already given, establishing } \overline{AB} \cong \overline{CD} \text{ satisfies condition 2.} \ &\text{Therefore, the required additional piece of information is } \overline{AB} \cong \overline{CD}. \end{aligned} $$

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

(B) $\overline{AC} \cong \overline{BD}$

(C) $\overline{AB} \perp \overline{BD}$

(D) $\overline{CD} \perp \overline{BD}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Parallelogram Proof Theorems" ] }

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QUESTION IMAGE

Question

Explanation:

Analyze the given geometric properties

Evaluate the sufficient conditions for a parallelogram

Match with the given options

Answer: