lines $l_1$ and $l_2$ are parallel lines. determine the measures of $\angle 1$ through $\angle 12$.

$m\angle 2 = 59^\circ$
$m\angle 3 = 64^\circ$
$m\angle 4 = 64^\circ$
$m\angle 5 = 57^\circ$
$m\angle 6 = \square^\circ$

Question

lines $l_1$ and $l_2$ are parallel lines. determine the measures of $\angle 1$ through $\angle 12$.

$m\angle 2 = 59^\circ$
$m\angle 3 = 64^\circ$
$m\angle 4 = 64^\circ$
$m\angle 5 = 57^\circ$
$m\angle 6 = \square^\circ$

Accepted Answer

{ "quality": "clear", "question_count": 12, "skills_matched": [ "step_cot", "table_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Parallel Lines Cut by a Transversal", "Vertical Angles", "Corresponding Angles", "Triangle Angle Sum Theorem", "Supplementary Angles" ], "new_concepts": [], "current_concepts": [ "Parallel Lines Cut by a Transversal", "Vertical Angles", "Corresponding Angles", "Triangle Angle Sum Theorem", "Supplementary Angles" ] } ## Find angle 6 and its vertical counterpart Using the *Vertical Angles* knowledge point $$ \begin{aligned} &m\angle 6 = 180^\circ - 64^\circ = 116^\circ\ &m\angle 4 = m\angle 64^\circ = 64^\circ \end{aligned} $$ ## Find angle 3 and angle 1 using parallel lines Using the *Parallel Lines Cut by a Transversal* and *Corresponding Angles* knowledge points $$ \begin{aligned} &m\angle 3 = m\angle 64^\circ = 64^\circ\ &m\angle 1 = m\angle 3 = 64^\circ \end{aligned} $$ ## Find angle 2 and angle 5 using the triangle sum Using the *Triangle Angle Sum Theorem* and *Supplementary Angles* knowledge points $$ \begin{aligned} &\text{Triangle interior angles: } 64^\circ, 59^\circ, m\angle 7\ &m\angle 7 = 180^\circ - (64^\circ + 59^\circ) = 57^\circ\ &m\angle 2 = 59^\circ\ &m\angle 5 = m\angle 7 = 57^\circ \end{aligned} $$ ## Find remaining angles on line 1 and line 2 Using the *Supplementary Angles* and *Vertical Angles* knowledge points $$ \begin{aligned} &m\angle 9 = 180^\circ - m\angle 6 = 64^\circ\ &m\angle 10 = m\angle 6 = 116^\circ\ &m\angle 8 = 180^\circ - m\angle 7 = 123^\circ\ &m\angle 11 = m\angle 8 = 123^\circ\ &m\angle 12 = m\angle 7 = 57^\circ \end{aligned} $$ | No. | Answer | |---|---| | 1 | $64^\circ$ | | 2 | $59^\circ$ | | 3 | $64^\circ$ | | 4 | $64^\circ$ | | 5 | $57^\circ$ | | 6 | $116^\circ$ | | 7 | $57^\circ$ | | 8 | $123^\circ$ | | 9 | $64^\circ$ | | 10 | $116^\circ$ | | 11 | $123^\circ$ | | 12 | $57^\circ$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Parallel Lines Cut by a Transversal" ] }

No.	Answer
2	\(59^\circ\)
3	\(64^\circ\)
4	\(64^\circ\)
5	\(57^\circ\)
6	\(116^\circ\)
7	\(57^\circ\)
8	\(123^\circ\)
9	\(64^\circ\)
10	\(116^\circ\)
11	\(123^\circ\)
12	\(57^\circ\)

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QUESTION IMAGE

Question

Explanation:

Find angle 6 and its vertical counterpart

Find angle 3 and angle 1 using parallel lines

Find angle 2 and angle 5 using the triangle sum

Find remaining angles on line 1 and line 2

Answer: