Question

in the diagram of circle c, \\(m\angle vwx\\) is \\(43^\circ\\). what is \\(m\widehat{ux}\\)?

\\(39^\circ\\)
\\(41^\circ\\)
\\(78^\circ\\)
\\(82^\circ\\)

Explanation:

<pre_analysis>
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"new_concepts": [],
"current_concepts": [
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"Arc Measure"
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</pre_analysis>

<reasoning>

Identify the given values and theorem

\[

$$\begin{aligned} &m\angle VWX = 43^\circ \\ &m\widehat{UY} = 128^\circ \\ &m\angle VWX = \frac{1}{2}(m\widehat{UY} - m\widehat{VX}) \end{aligned}$$

\]

Substitute the known values into the equation

\[
43^\circ = \frac{1}{2}(128^\circ - m\widehat{VX})
\]

Solve for the unknown arc measure

\[

$$\begin{aligned} &86^\circ = 128^\circ - m\widehat{VX} \\ &m\widehat{VX} = 128^\circ - 86^\circ \\ &m\widehat{VX} = 42^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 39°</mcq-option>
<mcq-option>(B) 41°</mcq-option>
<mcq-option>(C) 78°</mcq-option>
<mcq-correct>(D) 82°</mcq-correct>
</answer>

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

Answer:

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"new_concepts": [],
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</pre_analysis>

<reasoning>

Identify the given values and theorem

\[

$$\begin{aligned} &m\angle VWX = 43^\circ \\ &m\widehat{UY} = 128^\circ \\ &m\angle VWX = \frac{1}{2}(m\widehat{UY} - m\widehat{VX}) \end{aligned}$$

\]

Substitute the known values into the equation

\[
43^\circ = \frac{1}{2}(128^\circ - m\widehat{VX})
\]

Solve for the unknown arc measure

\[

$$\begin{aligned} &86^\circ = 128^\circ - m\widehat{VX} \\ &m\widehat{VX} = 128^\circ - 86^\circ \\ &m\widehat{VX} = 42^\circ \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) 39°</mcq-option>
<mcq-option>(B) 41°</mcq-option>
<mcq-option>(C) 78°</mcq-option>
<mcq-correct>(D) 82°</mcq-correct>
</answer>

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"question_type": "Multiple Choice",
"knowledge_point": [
"Mathematics",
"Geometry",
"Secant-Secant Angle Theorem"
]
}
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