Question

g.gsr.8.1 (mc)
\\(\angle acb\\) is a secant angle of circle f. \\(m\widehat{ab} = 170^\circ\\) and \\(m\widehat{de} = 40^\circ\\). find the measure of \\(\angle acb\\).

Explanation:

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

<reasoning>

State the formula for an angle formed by two secants intersecting outside a circle

\[
m\angle ACB = \frac{1}{2}(m\widehat{AB} - m\widehat{DE})
\]

Substitute the given arc measures into the formula

\[

$$\begin{aligned} m\widehat{AB} &= 170^\circ \\ m\widehat{DE} &= 40^\circ \\ m\angle ACB &= \frac{1}{2}(170^\circ - 40^\circ) \end{aligned}$$

\]

Calculate the final angle measure

\[

$$\begin{aligned} m\angle ACB &= \frac{1}{2}(130^\circ) \\ m\angle ACB &= 65^\circ \end{aligned}$$

\]
</reasoning>

<answer>
\(65^\circ\)
</answer>

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

Answer:

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

<reasoning>

State the formula for an angle formed by two secants intersecting outside a circle

\[
m\angle ACB = \frac{1}{2}(m\widehat{AB} - m\widehat{DE})
\]

Substitute the given arc measures into the formula

\[

$$\begin{aligned} m\widehat{AB} &= 170^\circ \\ m\widehat{DE} &= 40^\circ \\ m\angle ACB &= \frac{1}{2}(170^\circ - 40^\circ) \end{aligned}$$

\]

Calculate the final angle measure

\[

$$\begin{aligned} m\angle ACB &= \frac{1}{2}(130^\circ) \\ m\angle ACB &= 65^\circ \end{aligned}$$

\]
</reasoning>

<answer>
\(65^\circ\)
</answer>

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