carmen is designing an intersection of the rail line and four streets. she wants to know which streets are parallel.
which streets are parallel? check all that apply.
$c \parallel d$
$c \parallel e$
$c \parallel f$
$d \parallel e$
$d \parallel f$
$e \parallel f$

Question

Accepted Answer

{ "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", "Transversal Lines", "Vertical Angles", "Linear Pair" ], "new_concepts": [], "current_concepts": [ "Alternate Interior Angles", "Corresponding Angles", "Consecutive Interior Angles", "Alternate Exterior Angles", "Transversal Lines", "Vertical Angles", "Linear Pair", "Parallel Line Proofs" ] } ## Find corresponding interior angles for each line with transversal $t$ $$ \begin{aligned} &\theta_c = 180^\circ - 114^\circ = 66^\circ \quad \text{(interior angle, bottom-right of intersection with } c\text{)} \ &\theta_d = 68^\circ \quad \text{(interior angle, bottom-right of intersection with } d\text{)} \ &\theta_e = 180^\circ - 112^\circ = 68^\circ \quad \text{(interior angle, bottom-right of intersection with } e\text{)} \ &\theta_f = 66^\circ \quad \text{(interior angle, bottom-right of intersection with } f\text{)} \end{aligned} $$ ## Compare angles to identify parallel lines $$ \begin{aligned} &\theta_c = \theta_f = 66^\circ \implies c \parallel f \ &\theta_d = \theta_e = 68^\circ \implies d \parallel e \end{aligned} $$

c || d

c || e

c || f

d || e

d || f

e || f

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

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Question

Explanation:

Find corresponding interior angles for each line with transversal \(t\)

Compare angles to identify parallel lines

Answer: