Question

in the diagram of circle a, what is \\(m\angle lmn\\)?

\\(75^\circ\\)
\\(90^\circ\\)
\\(120^\circ\\)
\\(135^\circ\\)

Explanation:

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"current_concepts": [
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</pre_analysis>

<reasoning>

Calculate the measure of the intercepted minor arc

\[

$$\begin{aligned} \text{Major Arc Measure} &= 270^\circ \\ \text{Minor Arc Measure} &= 360^\circ - 270^\circ = 90^\circ \end{aligned}$$

\]

Apply the Circumscribed Angle Theorem

\[

$$\begin{aligned} m\angle LMN + \text{Minor Arc Measure} &= 180^\circ \\ m\angle LMN + 90^\circ &= 180^\circ \\ m\angle LMN &= 90^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(75^\circ\)</mcq-option>
<mcq-correct>(B) \(90^\circ\)</mcq-correct>
<mcq-option>(C) \(120^\circ\)</mcq-option>
<mcq-option>(D) \(135^\circ\)</mcq-option>
</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Circumscribed Angle Theorem"
]
}
</post_analysis>

Answer:

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</pre_analysis>

<reasoning>

Calculate the measure of the intercepted minor arc

\[

$$\begin{aligned} \text{Major Arc Measure} &= 270^\circ \\ \text{Minor Arc Measure} &= 360^\circ - 270^\circ = 90^\circ \end{aligned}$$

\]

Apply the Circumscribed Angle Theorem

\[

$$\begin{aligned} m\angle LMN + \text{Minor Arc Measure} &= 180^\circ \\ m\angle LMN + 90^\circ &= 180^\circ \\ m\angle LMN &= 90^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(75^\circ\)</mcq-option>
<mcq-correct>(B) \(90^\circ\)</mcq-correct>
<mcq-option>(C) \(120^\circ\)</mcq-option>
<mcq-option>(D) \(135^\circ\)</mcq-option>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
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