Question

which arc is congruent to \\(\widehat{eh}\\)?

\\(\bigcirc\\) \\(\widehat{gh}\\)
\\(\bigcirc\\) \\(\widehat{fh}\\)
\\(\bigcirc\\) \\(\widehat{ge}\\)
\\(\bigcirc\\) \\(\widehat{fg}\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Central Angle Theorem",
"Congruent Arcs",
"Circle Geometry"
],
"new_concepts": [],
"current_concepts": [
"Central Angle Theorem",
"Congruent Arcs",
"Circle Geometry"
]
}
</pre_analysis>

<reasoning>

Find the measure of the central angle for arc EH

\[

$$\begin{aligned} &FH \text{ is a line segment passing through the center } D \text{, so } \angle FDH = 180^\circ.\\ &m\angle EDF = 55^\circ\\ &m\angle EDH = 180^\circ - m\angle EDF = 180^\circ - 55^\circ = 125^\circ\\ &m\widehat{EH} = m\angle EDH = 125^\circ \end{aligned}$$

\]

Find the measures of the other central angles and arcs

\[

$$\begin{aligned} &m\angle FDG = 70^\circ \implies m\widehat{FG} = 70^\circ\\ &m\angle GDH = 110^\circ \implies m\widehat{GH} = 110^\circ\\ &m\angle EDG = m\angle EDF + m\angle FDG = 55^\circ + 70^\circ = 125^\circ\\ &m\widehat{GE} = m\angle EDG = 125^\circ \end{aligned}$$

\]

Compare arc measures to find the congruent arc

\[

$$\begin{aligned} &m\widehat{EH} = 125^\circ\\ &m\widehat{GE} = 125^\circ\\ &m\widehat{EH} = m\widehat{GE} \implies \widehat{EH} \cong \widehat{GE} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\widehat{GH}\)</mcq-option>
<mcq-option>(B) \(\widehat{FH}\)</mcq-option>
<mcq-correct>(C) \(\widehat{GE}\)</mcq-correct>
<mcq-option>(D) \(\widehat{FG}\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Circle Geometry"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"multiple_choice_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Central Angle Theorem",
"Congruent Arcs",
"Circle Geometry"
],
"new_concepts": [],
"current_concepts": [
"Central Angle Theorem",
"Congruent Arcs",
"Circle Geometry"
]
}
</pre_analysis>

<reasoning>

Find the measure of the central angle for arc EH

\[

$$\begin{aligned} &FH \text{ is a line segment passing through the center } D \text{, so } \angle FDH = 180^\circ.\\ &m\angle EDF = 55^\circ\\ &m\angle EDH = 180^\circ - m\angle EDF = 180^\circ - 55^\circ = 125^\circ\\ &m\widehat{EH} = m\angle EDH = 125^\circ \end{aligned}$$

\]

Find the measures of the other central angles and arcs

\[

$$\begin{aligned} &m\angle FDG = 70^\circ \implies m\widehat{FG} = 70^\circ\\ &m\angle GDH = 110^\circ \implies m\widehat{GH} = 110^\circ\\ &m\angle EDG = m\angle EDF + m\angle FDG = 55^\circ + 70^\circ = 125^\circ\\ &m\widehat{GE} = m\angle EDG = 125^\circ \end{aligned}$$

\]

Compare arc measures to find the congruent arc

\[

$$\begin{aligned} &m\widehat{EH} = 125^\circ\\ &m\widehat{GE} = 125^\circ\\ &m\widehat{EH} = m\widehat{GE} \implies \widehat{EH} \cong \widehat{GE} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(\widehat{GH}\)</mcq-option>
<mcq-option>(B) \(\widehat{FH}\)</mcq-option>
<mcq-correct>(C) \(\widehat{GE}\)</mcq-correct>
<mcq-option>(D) \(\widehat{FG}\)</mcq-option>
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Circle Geometry"
]
}
</post_analysis>