efgh is a parallelogram. select all statements that must be true.

efgh is a rectangle.
$\delta hef \cong \delta gfh$
$\delta hef \cong \delta fgh$
$\overline{ed} \cong \overline{hd} \cong \overline{dg} \cong \overline{df}$
$\delta edh \cong \delta hdg$

Question

efgh is a parallelogram. select all statements that must be true.

efgh is a rectangle.
$\delta hef \cong \delta gfh$
$\delta hef \cong \delta fgh$
$\overline{ed} \cong \overline{hd} \cong \overline{dg} \cong \overline{df}$
$\delta edh \cong \delta hdg$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Parallelogram Properties", "Rectangle Properties" ], "new_concepts": [], "current_concepts": [ "Parallelogram Properties", "Rectangle Properties", "Triangle Congruence" ] } ## Analyze the given geometric properties $$ \begin{aligned} &\text{Given: } EFGH \text{ is a parallelogram.} \ &\text{Given from diagram: } \angle FEH = 90^\circ \text{ (indicated by the right-angle symbol at vertex } E\text{).} \ &\text{Since a parallelogram with one right angle is a rectangle:} \ &EFGH \text{ is a rectangle.} \end{aligned} $$ ## Evaluate each statement based on rectangle properties $$ \begin{aligned} &1. \text{ "EFGH is a rectangle": True (proven above).} \ &2. \text{ "$\triangle HEF \cong \triangle GFH$":} \ &\quad HE = GF \text{ (opposite sides of rectangle)} \ &\quad EF = FH \text{ is false (FH is a diagonal, EF is a side).} \implies \text{False.} \ &3. \text{ "$\triangle HEF \cong \triangle FGH$":} \ &\quad HE = GF, \quad EF = HG, \quad HF = FH \implies \text{True by SSS.} \ &4. \text{ "$\overline{ED} \cong \overline{HD} \cong \overline{DG} \cong \overline{DF}$":} \ &\quad \text{Diagonals of a rectangle are equal and bisect each other:} \ &\quad EG = HF \implies ED = DG = HD = DF \implies \text{True.} \ &5. \text{ "$\triangle EDH \cong \triangle HDG$":} \ &\quad \text{Only true if } EH = HG \text{ (which is a square, not necessarily true for all rectangles).} \implies \text{False.} \end{aligned} $$

EFGH is a rectangle.

$\triangle HEF \cong \triangle GFH$

$\triangle HEF \cong \triangle FGH$

$\overline{ED} \cong \overline{HD} \cong \overline{DG} \cong \overline{DF}$

$\triangle EDH \cong \triangle HDG$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Rectangle Properties" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Analyze the given geometric properties

Evaluate each statement based on rectangle properties

Answer: