in the figure, m is parallel to n and $m\angle 4 = 128^{\circ}$. find the measures of the other angles.

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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Supplementary Angles" ], "new_concepts": [ "Parallel Lines Cut by a Transversal", "Vertical Angles", "Corresponding Angles" ], "current_concepts": [ "Supplementary Angles", "Parallel Lines Cut by a Transversal", "Vertical Angles", "Corresponding Angles" ] } ## Find supplementary angles to angle 4 Using the *Supplementary Angles* knowledge point $$ \begin{aligned} &m\angle 3 + m\angle 4 = 180^\circ \implies m\angle 3 = 180^\circ - 128^\circ = 52^\circ\ &m\angle 5 + m\angle 4 = 180^\circ \implies m\angle 5 = 180^\circ - 128^\circ = 52^\circ \end{aligned} $$ ## Find vertical angle to angle 4 We use the vertical angles theorem to find the measure of angle 6. Since vertical angles are equal: $$ m\angle 6 = m\angle 4 = 128^\circ $$ ## Find corresponding angles on line n Since line $m$ is parallel to line $n$, corresponding angles are equal. We match each angle on line $m$ with its corresponding angle on line $n$: $$ \begin{aligned} &m\angle 1 = m\angle 3 = 52^\circ\ &m\angle 2 = m\angle 4 = 128^\circ\ &m\angle 7 = m\angle 5 = 52^\circ\ &m\angle 8 = m\angle 6 = 128^\circ \end{aligned} $$ $$ \begin{aligned} m\angle 1 &= 52^\circ \ m\angle 2 &= 128^\circ \ m\angle 3 &= 52^\circ \ m\angle 5 &= 52^\circ \ m\angle 6 &= 128^\circ \ m\angle 7 &= 52^\circ \ m\angle 8 &= 128^\circ \end{aligned} $$ { "subject": "Mathematics", "question_type": "Free Response", "knowledge_point": [ "Mathematics", "Geometry", "Parallel Lines Cut by a Transversal" ] }

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Explanation:

Find supplementary angles to angle 4

Find vertical angle to angle 4

Find corresponding angles on line n

Answer: