in the diagram below, $fg = 10$ and $hj = 10$. which additional facts would guarantee that $fghj$ is a parallelogram? check all that apply.

a. $fj = 10, gh = 15$
b. $fj = 15, gh = 15$
c. $gh = 10$
d. $\overline{fg}$ is parallel to $\overline{hj}$.

Question

in the diagram below, $fg = 10$ and $hj = 10$. which additional facts would guarantee that $fghj$ is a parallelogram? check all that apply.

a. $fj = 10, gh = 15$
b. $fj = 15, gh = 15$
c. $gh = 10$
d. $\overline{fg}$ is parallel to $\overline{hj}$.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Parallelogram Converse Theorems" ], "new_concepts": [], "current_concepts": [ "Parallelogram Converse Theorems", "Parallelogram Properties" ] } ## Analyze the given conditions $$ \begin{aligned} &FG = 10 \ &HJ = 10 \ &FG = HJ = 10 \quad (\text{One pair of opposite sides is congruent}) \end{aligned} $$ ## Evaluate the converse theorems for parallelograms $$ \begin{aligned} &\text{Theorem 1: Both pairs of opposite sides are congruent.} \ &\text{Theorem 2: One pair of opposite sides is both parallel and congruent.} \end{aligned} $$ ## Test each option $$ \begin{aligned} &\text{A. } FJ = 10, GH = 15 \implies FJ eq GH \quad (\text{Opposite sides not congruent, not guaranteed}) \ &\text{B. } FJ = 15, GH = 15 \implies FJ = GH = 15 \quad (\text{Both pairs of opposite sides congruent} \implies \text{Guaranteed}) \ &\text{C. } GH = 10 \implies \text{Only gives } GH = 10 \text{ without knowing } FJ \quad (\text{Not guaranteed}) \ &\text{D. } \overline{FG} \parallel \overline{HJ} \implies \text{One pair of opposite sides is both parallel and congruent} \implies \text{Guaranteed} \end{aligned} $$

A. $FJ = 10, GH = 15$

B. $FJ = 15, GH = 15$

C. $GH = 10$

D. $\overline{FG}$ is parallel to $\overline{HJ}$

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

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QUESTION IMAGE

Question

Explanation:

Analyze the given conditions

Evaluate the converse theorems for parallelograms

Test each option

Answer: