Question

select the term that describes the intersection or union. use uppercase letters to name the figure.

a. \\(ab \cup bd\\)

a. \\(\overline{db} \cap \overline{bc}\\)

a. \\(\overline{ab} \cup \overline{bd}\\)

Explanation:

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"Rays and Segments",
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"Intersection and Union"
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</pre_analysis>

<reasoning>

Analyze the first operation

\[
\overrightarrow{AB} \cup \overrightarrow{BD}
\]
The ray \(\overrightarrow{AB}\) starts at \(A\) and extends infinitely to the right through \(B, C, D\).
The ray \(\overrightarrow{BD}\) starts at \(B\) and extends infinitely to the right through \(C, D\).
Their union is the larger set, which is the ray starting at \(A\) and extending infinitely to the right:
\[
\overrightarrow{AB} \cup \overrightarrow{BD} = \overrightarrow{AB} \quad (\text{or } \overrightarrow{AC}, \overrightarrow{AD})
\]
The term is "Ray" and the name is "AB".

Analyze the second operation

\[
\overrightarrow{DB} \cap \overline{BC}
\]
The ray \(\overrightarrow{DB}\) starts at \(D\) and extends infinitely to the left through \(C, B, A\).
The segment \(\overline{BC}\) consists of all points between \(B\) and \(C\) inclusive.
Since \(\overline{BC}\) lies entirely within the ray \(\overrightarrow{DB}\), their intersection is the segment itself:
\[
\overrightarrow{DB} \cap \overline{BC} = \overline{BC}
\]
The term is "Segment" and the name is "BC".

Analyze the third operation

\[
\overline{AB} \cup \overline{BD}
\]
The segment \(\overline{AB}\) consists of points from \(A\) to \(B\).
The segment \(\overline{BD}\) consists of points from \(B\) to \(D\).
Since they share the endpoint \(B\), their union is the single continuous segment from \(A\) to \(D\):
\[
\overline{AB} \cup \overline{BD} = \overline{AD}
\]
The term is "Segment" and the name is "AD".
</reasoning>

<answer>

Question 1

• Term: Ray
• Name: AB

Question 2

• Term: Segment
• Name: BC

Question 3

• Term: Segment
• Name: AD

</answer>

<post_analysis>
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"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Geometric Set Operations"
]
}
</post_analysis>

Answer:

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"Intersection and Union"
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</pre_analysis>

<reasoning>

Analyze the first operation

\[
\overrightarrow{AB} \cup \overrightarrow{BD}
\]
The ray \(\overrightarrow{AB}\) starts at \(A\) and extends infinitely to the right through \(B, C, D\).
The ray \(\overrightarrow{BD}\) starts at \(B\) and extends infinitely to the right through \(C, D\).
Their union is the larger set, which is the ray starting at \(A\) and extending infinitely to the right:
\[
\overrightarrow{AB} \cup \overrightarrow{BD} = \overrightarrow{AB} \quad (\text{or } \overrightarrow{AC}, \overrightarrow{AD})
\]
The term is "Ray" and the name is "AB".

Analyze the second operation

\[
\overrightarrow{DB} \cap \overline{BC}
\]
The ray \(\overrightarrow{DB}\) starts at \(D\) and extends infinitely to the left through \(C, B, A\).
The segment \(\overline{BC}\) consists of all points between \(B\) and \(C\) inclusive.
Since \(\overline{BC}\) lies entirely within the ray \(\overrightarrow{DB}\), their intersection is the segment itself:
\[
\overrightarrow{DB} \cap \overline{BC} = \overline{BC}
\]
The term is "Segment" and the name is "BC".

Analyze the third operation

\[
\overline{AB} \cup \overline{BD}
\]
The segment \(\overline{AB}\) consists of points from \(A\) to \(B\).
The segment \(\overline{BD}\) consists of points from \(B\) to \(D\).
Since they share the endpoint \(B\), their union is the single continuous segment from \(A\) to \(D\):
\[
\overline{AB} \cup \overline{BD} = \overline{AD}
\]
The term is "Segment" and the name is "AD".
</reasoning>

<answer>

Question 1

• Term: Ray
• Name: AB

Question 2

• Term: Segment
• Name: BC

Question 3

• Term: Segment
• Name: AD

</answer>

<post_analysis>
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"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Geometry",
"Geometric Set Operations"
]
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</post_analysis>