if quadrilateral $pqrs$ is a parallelogram, which statement is always true?

$m\angle p = m\angle q$
$\overline{pq}$ is parallel to $\overline{qr}$
$\angle r$ is an acute angle
$\overline{ps} \cong \overline{qr}$

Question

if quadrilateral $pqrs$ is a parallelogram, which statement is always true?

$m\angle p = m\angle q$
$\overline{pq}$ is parallel to $\overline{qr}$
$\angle r$ is an acute angle
$\overline{ps} \cong \overline{qr}$

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" ], "new_concepts": [], "current_concepts": [ "Parallelogram Properties" ] } ## Analyze quadrilateral properties $$ \begin{aligned} &\text{Let } PQRS \text{ be a parallelogram with vertices in order } P, Q, R, S.\ &\text{Opposite sides: } \overline{PQ} \parallel \overline{SR} \text{ and } \overline{PS} \parallel \overline{QR}.\ &\text{Opposite side lengths: } PQ = SR \text{ and } PS = QR \implies \overline{PS} \cong \overline{QR}.\ &\text{Opposite angles: } m\angle P = m\angle R \text{ and } m\angle Q = m\angle S.\ &\text{Consecutive angles: } m\angle P + m\angle Q = 180^\circ. \end{aligned} $$ ## Evaluate the given statements $$ \begin{aligned} &1. \ m\angle P = m\angle Q \quad (\text{False in general, only true if rectangle})\ &2. \ \overline{PQ} \parallel \overline{QR} \quad (\text{False, they are intersecting consecutive sides})\ &3. \ \angle R \text{ is an acute angle} \quad (\text{False in general, can be right or obtuse})\ &4. \ \overline{PS} \cong \overline{QR} \quad (\text{Always true, opposite sides of a parallelogram are congruent}) \end{aligned} $$

(A) $m\angle P = m\angle Q$

(B) $\overline{PQ}$ is parallel to $\overline{QR}$

(C) $\angle R$ is an acute angle

(D) $\overline{PS} \cong \overline{QR}$

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

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Question

Explanation:

Analyze quadrilateral properties

Evaluate the given statements

Answer: