consider transversal $t$ cutting lines $p$ and $q$.

which statement allows carmen to conclude that line $p$ and line $q$ are parallel?

$\bigcirc\ \angle 2 \cong \angle 6$
$\bigcirc\ \angle 1 \cong \angle 4$
$\bigcirc\ m\angle 1 + m\angle 2 = 180^\circ$
$\bigcirc\ m\angle 5 + m\angle 7 = 180^\circ$

Question

consider transversal $t$ cutting lines $p$ and $q$.

which statement allows carmen to conclude that line $p$ and line $q$ are parallel?

$\bigcirc\ \angle 2 \cong \angle 6$
$\bigcirc\ \angle 1 \cong \angle 4$
$\bigcirc\ m\angle 1 + m\angle 2 = 180^\circ$
$\bigcirc\ m\angle 5 + m\angle 7 = 180^\circ$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Corresponding Angles", "Vertical Angles", "Linear Pair" ], "new_concepts": [ "Parallel Line Proofs" ], "current_concepts": [ "Corresponding Angles", "Vertical Angles", "Linear Pair", "Parallel Line Proofs" ] } ## Identify angle relationships between lines $p$ and $q$ $$ \begin{aligned} &\angle 2 \text{ and } \angle 6 \text{ are corresponding angles.} \ &\angle 1 \text{ and } \angle 4 \text{ are vertical angles on line } p. \ &\angle 1 \text{ and } \angle 2 \text{ form a linear pair on line } p. \ &\angle 5 \text{ and } \angle 7 \text{ form a linear pair on line } q. \end{aligned} $$ ## Apply parallel line converse theorems $$ \begin{aligned} &\angle 2 \cong \angle 6 \implies p \parallel q \quad (\text{Corresponding Angles Converse}) \ &\angle 1 \cong \angle 4 \implies \text{always true, does not prove } p \parallel q \ &m\angle 1 + m\angle 2 = 180^\circ \implies \text{always true, does not prove } p \parallel q \ &m\angle 5 + m\angle 7 = 180^\circ \implies \text{always true, does not prove } p \parallel q \end{aligned} $$

(A) $\angle 2 \cong \angle 6$

(B) $\angle 1 \cong \angle 4$

(C) $m\angle 1 + m\angle 2 = 180^\circ$

(D) $m\angle 5 + m\angle 7 = 180^\circ$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Parallel Line Proofs" ] }

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Question

Explanation:

Identify angle relationships between lines \(p\) and \(q\)

Apply parallel line converse theorems

Answer: