find the parallel lines.

which pair of lines are parallel?
$p \parallel s$
$q \parallel r$
$p \parallel q$
$r \parallel s$

Question

find the parallel lines.

which pair of lines are parallel?
$p \parallel s$
$q \parallel r$
$p \parallel q$
$r \parallel s$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Linear Pair", "Corresponding Angles" ], "new_concepts": [], "current_concepts": [ "Linear Pair", "Corresponding Angles", "Transversal Lines" ] } ## Find corresponding angles for each line with transversal $t$ $$ \begin{aligned} &\theta_p = 180^\circ - 116^\circ = 64^\circ \quad \text{(interior angle, top-right)} \ &\theta_q = 66^\circ \quad \text{(interior angle, top-right)} \ &\theta_r = 114^\circ \quad \text{(interior angle, bottom-left)} \ &\theta_s = 180^\circ - 74^\circ = 106^\circ \quad \text{(interior angle, bottom-left)} \end{aligned} $$ ## Compare corresponding angles to find parallel lines \[ \begin{aligned} &\text{For line } p \text{ and line } r \text{ (using bottom-left interior angles):} \ &\theta_{p,\text{bottom-left}} = 180^\circ - 64^\circ = 116^\circ \ &\theta_{r,\text{bottom-left}} = 114^\circ \ &116^\circ eq 114^\circ \implies p ot\parallel r \ \ &\text{For line } q \text{ and line } s \text{ (using bottom-left interior angles):} \ &\theta_{q,\text{bottom-left}} = 180^\circ - 66^\circ = 114^\circ \ &\theta_{s,\text{bottom-left}} = 180^\circ - 106^\circ = 74^\circ \ &114^\circ eq 74^\circ \implies q ot\parallel s \ \ &\text{For line } r \text{ and line } s \text{ (using bottom-left interior angles):} \ &\theta_{r,\text{bottom-left}} = 114^\circ \ &\theta_{s,\text{bottom-left}} = 180^\circ - 74^\circ = 106^\circ \ &114^\circ eq 106^\circ \implies r ot\parallel s \ \ &\text{For line } p \text{ and line } s \text{ (using top-right exterior/interior angles):} \ &\text{Angle above } p \text{ is } 116^\circ. \ &\text{Angle below } s \text{ is } 74^\circ \implies \text{Angle above } s \text{ is } 180^\circ - 74^\circ = 106^\circ. \ &116^\circ eq 106^\circ \implies p ot\parallel s \ \ &\text{For line } q \text{ and line } r \text{ (using consecutive interior angles):} \ &\text{Interior angle of } q \text{ (top-right) } = 66^\circ \ &\text{Interior angle of } r \text{ (bottom-left) } = 114^\circ \ &\text{Consecutive interior sum: } 66^\circ + 114^\circ = 180^\circ \implies q \parallel r \end{aligned} \]

$p \parallel s$

$q \parallel r$

$p \parallel q$

$r \parallel s$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Corresponding Angles" ] }

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QUESTION IMAGE

Question

Explanation:

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

Compare corresponding angles to find parallel lines

Answer: