find the measure of angle r. ______^
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{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": true, "concepts_used": [ "Parallel Lines Cut by a Transversal", "Alternate Interior Angles", "Linear Pair", "Triangle Angle Sum" ], "new_concepts": [], "current_concepts": [ "Parallel Lines Cut by a Transversal", "Alternate Interior Angles", "Linear Pair", "Triangle Angle Sum" ] } ## Identify the geometric structure Using the *Parallel Lines Cut by a Transversal* knowledge point The arrows on the two horizontal lines indicate they are parallel. The two intersecting lines form a triangle with the parallel lines. ## Find the interior angles of the triangle Using the *Linear Pair* and *Alternate Interior Angles* knowledge points $$ \begin{aligned} &\text{Let the interior angles of the triangle be } a, b, c.\ &\text{At the bottom-right vertex, the exterior angle is } 142^\circ.\ &\text{The adjacent interior angle } a = 180^\circ - 142^\circ = 38^\circ.\ &\text{At the top vertex, the angle alternate interior to the bottom-left interior angle is } 41^\circ.\ &\text{Thus, the bottom-left interior angle } b = 41^\circ. \end{aligned} $$ ## Calculate the third interior angle Using the *Triangle Angle Sum* knowledge point $$ \begin{aligned} &c = 180^\circ - (a + b)\ &c = 180^\circ - (38^\circ + 41^\circ) = 101^\circ \end{aligned} $$ ## Solve for angle r Using the *Linear Pair* knowledge point $$ \begin{aligned} &r + 41^\circ + c = 180^\circ \text{ along the top straight line}\ &r = 180^\circ - 41^\circ - 101^\circ = 38^\circ \end{aligned} $$ Find the measure of angle r. 38$^\circ$ { "elements": [ { "type": "line", "params": [[-5, 3], [5, 3]], "properties": { "strokeColor": "#583C87", "strokeWidth": 2 } }, { "type": "line", "params": [[-5, -2], [5, -2]], "properties": { "strokeColor": "#583C87", "strokeWidth": 2 } }, { "type": "line", "params": [[-4, -3], [2, 4]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "line", "params": [[-1, 4], [4, -3]], "properties": { "strokeColor": "#8C55F2", "strokeWidth": 2 } }, { "type": "point", "params": [[1.14, 3]], "properties": { "name": "r", "withLabel": true, "color": "#F2557F", "size": 3 } }, { "type": "text", "params": [0.5, 2.5, "41°"] }, { "type": "text", "params": [3.2, -1.5, "142°"] } ] } { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Parallel Lines Cut by a Transversal" ] }

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Question

Explanation:

Identify the geometric structure

Find the interior angles of the triangle

Calculate the third interior angle

Solve for angle r

Answer: