Question

the terminal angle of \\(m\angle def\\) is \\(139^\circ\\). what is the measure of its reference angle?

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": true,
"concepts_used": [
"Trigonometric Reference Angles"
],
"new_concepts": [],
"current_concepts": [
"Trigonometric Reference Angles"
]
}
</pre_analysis>

<reasoning>

Identify the quadrant of the given angle

\[
90^\circ < 139^\circ < 180^\circ \implies \text{Quadrant II}
\]

Apply the reference angle formula for Quadrant II

\[
\theta' = 180^\circ - \theta
\]
\[
\theta' = 180^\circ - 139^\circ
\]

Calculate the final measure

\[
\theta' = 41^\circ
\]
</reasoning>

<answer>
The terminal angle of m\(\angle\)DEF is \(139^\circ\). What is the measure of its reference angle? <blank>\(41^\circ\)</blank>
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[-5, 0], [5, 0]],
"properties": {
"strokeColor": "#cccccc",
"strokeWidth": 1,
"fixed": true
}
},
{
"type": "line",
"params": [[0, -5], [0, 5]],
"properties": {
"strokeColor": "#cccccc",
"strokeWidth": 1,
"fixed": true
}
},
{
"type": "circle",
"params": [[0, 0], 3],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2,
"fixed": true
}
},
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "E",
"color": "#583C87",
"size": 3,
"withLabel": true,
"labelPosition": "bottom"
}
},
{
"type": "point",
"params": [[3, 0]],
"properties": {
"name": "F",
"color": "#583C87",
"size": 3,
"withLabel": true,
"labelPosition": "right"
}
},
{
"type": "ray",
"params": [[0, 0], [4.5, 0]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 3
}
},
{
"type": "point",
"params": [[-2.264, 1.968]],
"properties": {
"name": "D",
"color": "#583C87",
"size": 3,
"withLabel": true,
"labelPosition": "left"
}
},
{
"type": "ray",
"params": [[0, 0], [-3.396, 2.952]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 3
}
},
{
"type": "angle",
"params": [[3, 0], [0, 0], [-2.264, 1.968]],
"properties": {
"radius": 0.8,
"fillColor": "rgba(140, 85, 242, 0.3)",
"strokeColor": "#8C55F2",
"withLabel": true,
"name": "139°"
}
},
{
"type": "angle",
"params": [[-2.264, 1.968], [0, 0], [-3, 0]],
"properties": {
"radius": 1.2,
"fillColor": "rgba(85, 221, 242, 0.3)",
"strokeColor": "#55DDF2",
"withLabel": true,
"name": "41°"
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Trigonometric Reference Angles"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": true,
"concepts_used": [
"Trigonometric Reference Angles"
],
"new_concepts": [],
"current_concepts": [
"Trigonometric Reference Angles"
]
}
</pre_analysis>

<reasoning>

Identify the quadrant of the given angle

\[
90^\circ < 139^\circ < 180^\circ \implies \text{Quadrant II}
\]

Apply the reference angle formula for Quadrant II

\[
\theta' = 180^\circ - \theta
\]
\[
\theta' = 180^\circ - 139^\circ
\]

Calculate the final measure

\[
\theta' = 41^\circ
\]
</reasoning>

<answer>
The terminal angle of m\(\angle\)DEF is \(139^\circ\). What is the measure of its reference angle? <blank>\(41^\circ\)</blank>
</answer>

<plot>
{
"elements": [
{
"type": "line",
"params": [[-5, 0], [5, 0]],
"properties": {
"strokeColor": "#cccccc",
"strokeWidth": 1,
"fixed": true
}
},
{
"type": "line",
"params": [[0, -5], [0, 5]],
"properties": {
"strokeColor": "#cccccc",
"strokeWidth": 1,
"fixed": true
}
},
{
"type": "circle",
"params": [[0, 0], 3],
"properties": {
"strokeColor": "#5583F2",
"strokeWidth": 2,
"fixed": true
}
},
{
"type": "point",
"params": [[0, 0]],
"properties": {
"name": "E",
"color": "#583C87",
"size": 3,
"withLabel": true,
"labelPosition": "bottom"
}
},
{
"type": "point",
"params": [[3, 0]],
"properties": {
"name": "F",
"color": "#583C87",
"size": 3,
"withLabel": true,
"labelPosition": "right"
}
},
{
"type": "ray",
"params": [[0, 0], [4.5, 0]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 3
}
},
{
"type": "point",
"params": [[-2.264, 1.968]],
"properties": {
"name": "D",
"color": "#583C87",
"size": 3,
"withLabel": true,
"labelPosition": "left"
}
},
{
"type": "ray",
"params": [[0, 0], [-3.396, 2.952]],
"properties": {
"strokeColor": "#F2557F",
"strokeWidth": 3
}
},
{
"type": "angle",
"params": [[3, 0], [0, 0], [-2.264, 1.968]],
"properties": {
"radius": 0.8,
"fillColor": "rgba(140, 85, 242, 0.3)",
"strokeColor": "#8C55F2",
"withLabel": true,
"name": "139°"
}
},
{
"type": "angle",
"params": [[-2.264, 1.968], [0, 0], [-3, 0]],
"properties": {
"radius": 1.2,
"fillColor": "rgba(85, 221, 242, 0.3)",
"strokeColor": "#55DDF2",
"withLabel": true,
"name": "41°"
}
}
]
}
</plot>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Trigonometric Reference Angles"
]
}
</post_analysis>