if $x = 10$, which lines are parallel? check all that apply.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Alternate Interior Angles", "Corresponding Angles", "Consecutive Interior Angles", "Alternate Exterior Angles" ], "new_concepts": [], "current_concepts": [ "Alternate Interior Angles", "Corresponding Angles", "Consecutive Interior Angles", "Alternate Exterior Angles", "Parallel Line Proofs" ] } ## Evaluate angle measures at each intersection for $x = 10$ $$ \begin{aligned} &\text{At line } b: \theta_b = 7(10) - 5 = 65^\circ \quad (\text{obtuse-side supplementary: } 180^\circ - 65^\circ = 115^\circ) \ &\text{At line } c: \theta_c = 10(10) + 15 = 115^\circ \ &\text{At line } d: \theta_d = 12(10) - 5 = 115^\circ \ &\text{At line } e: \theta_e = 8(10) - 25 = 55^\circ \quad (\text{obtuse-side supplementary: } 180^\circ - 55^\circ = 125^\circ) \end{aligned} $$ ## Compare corresponding angle positions along transversal $a$ $$ \begin{aligned} &\text{Obtuse angle at } b: 115^\circ \ &\text{Obtuse angle at } c: 115^\circ \ &\text{Obtuse angle at } d: 115^\circ \ &\text{Obtuse angle at } e: 125^\circ \end{aligned} $$ ## Identify parallel lines based on equal corresponding angles $$ \begin{aligned} &\theta_{\text{obtuse}, b} = \theta_{\text{obtuse}, c} = \theta_{\text{obtuse}, d} = 115^\circ \implies b \parallel c \parallel d \ &\theta_{\text{obtuse}, e} = 125^\circ eq 115^\circ \implies e \text{ is not parallel to } b, c, \text{ or } d \end{aligned} $$

(A) Lines $b$, $c$, and $d$ are parallel

(B) Lines $b$ and $c$ only

(C) Lines $c$ and $d$ only

(D) Lines $b$, $c$, $d$, and $e$ are parallel

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

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Question

Explanation:

Evaluate angle measures at each intersection for \(x = 10\)

Compare corresponding angle positions along transversal \(a\)

Identify parallel lines based on equal corresponding angles

Answer: