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

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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", "Interior Angles of a Triangle", "Angles on a Straight Line" ], "new_concepts": [], "current_concepts": [ "Parallel Lines Cut by a Transversal", "Vertical Angles", "Interior Angles of a Triangle", "Angles on a Straight Line" ] } ## Find angles at the intersection with transversal 1 Using the *Parallel Lines Cut by a Transversal* and *Vertical Angles* knowledge points $$ \begin{aligned} &\angle 10 = 69^\circ \quad (\text{given})\ &\angle 6 = 180^\circ - 69^\circ = 111^\circ\ &\angle 9 = \angle 10 = 69^\circ\ &\angle 6 = \angle 10 = 111^\circ \implies \angle 6 = 111^\circ\ &\angle 2 = \angle 10 = 69^\circ \quad (\text{corresponding angles})\ &\angle 4 = \angle 2 = 69^\circ \quad (\text{vertical angles})\ &\angle 1 = 180^\circ - \angle 2 = 111^\circ \quad (\text{angles on a straight line})\ &\angle 3 = \angle 1 = 111^\circ \quad (\text{vertical angles}) \end{aligned} $$ ## Find the interior angles of the triangle Using the *Interior Angles of a Triangle* and *Angles on a Straight Line* knowledge points $$ \begin{aligned} &\text{The triangle is formed by } L_1, L_2, \text{ and the two transversals.}\ &\text{The interior angle at the top vertex is } 57^\circ.\ &\text{The interior angle at the bottom-left vertex is } 69^\circ.\ &\text{Let the interior angle at the bottom-right vertex be } \angle 7.\ &\angle 7 = 180^\circ - (57^\circ + 69^\circ) = 54^\circ \end{aligned} $$ ## Find angles at the intersection with transversal 2 Using the *Angles on a Straight Line* and *Vertical Angles* knowledge points $$ \begin{aligned} &\angle 7 = 54^\circ\ &\angle 8 = 180^\circ - \angle 7 = 126^\circ\ &\angle 11 = \angle 8 = 126^\circ\ &\angle 12 = \angle 7 = 54^\circ \end{aligned} $$ ## Find the remaining angles on line 1 Using the *Parallel Lines Cut by a Transversal* and *Vertical Angles* knowledge points $$ \begin{aligned} &\angle 5 = \angle 7 = 54^\circ \quad (\text{alternate interior angles})\ &\angle 3 = 111^\circ \quad (\text{from Step 1}) \end{aligned} $$ | No. | Answer | |---|---| | 1 | $111^\circ$ | | 2 | $69^\circ$ | | 3 | $111^\circ$ | | 4 | $69^\circ$ | | 5 | $54^\circ$ | | 6 | $111^\circ$ | | 7 | $54^\circ$ | | 8 | $126^\circ$ | | 9 | $69^\circ$ | | 10 | $69^\circ$ | | 11 | $126^\circ$ | | 12 | $54^\circ$ | { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Parallel Lines Cut by a Transversal" ] }

No.	Answer
2	\(69^\circ\)
3	\(111^\circ\)
4	\(69^\circ\)
5	\(54^\circ\)
6	\(111^\circ\)
7	\(54^\circ\)
8	\(126^\circ\)
9	\(69^\circ\)
10	\(69^\circ\)
11	\(126^\circ\)
12	\(54^\circ\)

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Question

Explanation:

Find angles at the intersection with transversal 1

Find the interior angles of the triangle

Find angles at the intersection with transversal 2

Find the remaining angles on line 1

Answer: