Question

g.gsr.8.1 (mc)

\\(\angle bdc = 70^\circ\\) on circle a. find the measure of \\(\widehat{bc}\\).

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Inscribed Angle",
"Arc Measure"
],
"new_concepts": [],
"current_concepts": [
"Inscribed Angle",
"Arc Measure"
]
}
</pre_analysis>

<reasoning>

Identify the given angle and its intercepted arc

\[

$$\begin{aligned} &\angle BDC = 70^\circ \quad (\text{Inscribed angle on Circle } A) \\ &\text{Intercepted Arc} = \widehat{BC} \end{aligned}$$

\]

Apply the Inscribed Angle Theorem

\[

$$\begin{aligned} &m\angle BDC = \frac{1}{2} \cdot m\widehat{BC} \\ &70^\circ = \frac{1}{2} \cdot m\widehat{BC} \end{aligned}$$

\]

Solve for the measure of the arc

\[

$$\begin{aligned} &m\widehat{BC} = 2 \cdot 70^\circ \\ &m\widehat{BC} = 140^\circ \end{aligned}$$

\]
</reasoning>

<answer>
\(140^\circ\)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Inscribed Angle"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Inscribed Angle",
"Arc Measure"
],
"new_concepts": [],
"current_concepts": [
"Inscribed Angle",
"Arc Measure"
]
}
</pre_analysis>

<reasoning>

Identify the given angle and its intercepted arc

\[

$$\begin{aligned} &\angle BDC = 70^\circ \quad (\text{Inscribed angle on Circle } A) \\ &\text{Intercepted Arc} = \widehat{BC} \end{aligned}$$

\]

Apply the Inscribed Angle Theorem

\[

$$\begin{aligned} &m\angle BDC = \frac{1}{2} \cdot m\widehat{BC} \\ &70^\circ = \frac{1}{2} \cdot m\widehat{BC} \end{aligned}$$

\]

Solve for the measure of the arc

\[

$$\begin{aligned} &m\widehat{BC} = 2 \cdot 70^\circ \\ &m\widehat{BC} = 140^\circ \end{aligned}$$

\]
</reasoning>

<answer>
\(140^\circ\)
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Free Response",
"knowledge_point": [
"Mathematics",
"Geometry",
"Inscribed Angle"
]
}
</post_analysis>