Question

proofs are composed of statements and reasons. determine if each entry is a reason or a statement.

vertical angles of theorem
\\(\overleftrightarrow{pq}\\) bisects \\(\overline{rs}\\)
\\(\overline{ab} \cong \overline{bc}\\)
given
definition of midpoint
linear pair postulate
reflexive property
\\(m\angle abc = 90^{\circ}\\)
\\(\angle a\\) and \\(\angle d\\) are complementary angles
\\(\angle 1 \cong \angle 2\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"direct_cot",
"table_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Geometric Proofs",
"Mathematical Logic",
"Geometric Definitions",
"Postulates and Axioms"
],
"new_concepts": [],
"current_concepts": [
"Geometric Proofs",
"Mathematical Logic",
"Geometric Definitions",
"Postulates and Axioms"
]
}
</pre_analysis>

<reasoning>

Distinguish between statements and reasons in geometric proofs

A statement is a specific mathematical assertion or claim about a geometric figure (e.g., congruence, equality, bisecting relationships). A reason is a justification, definition, postulate, theorem, or property that supports why a statement is true.

Classify each given entry

• "Vertical Angles of Theorem" (likely a typo for "Vertical Angles Theorem"): Reason
• "\(\overleftrightarrow{PQ}\) bisects \(\overline{RS}\)": Statement
• "\(\overline{AB} \cong \overline{BC}\)": Statement
• "Given": Reason
• "Definition of Midpoint": Reason
• "Linear Pair Postulate": Reason
• "Reflexive Property": Reason
• "\(m\angle ABC = 90^\circ\)": Statement
• "\(\angle A\) and \(\angle D\) are complementary angles": Statement
• "\(\angle 1 \cong \angle 2\)": Statement

</reasoning>

<answer>

Entry	Classification
\(\overleftrightarrow{PQ}\) bisects \(\overline{RS}\)	Statement
\(\overline{AB} \cong \overline{BC}\)	Statement
Given	Reason
Definition of Midpoint	Reason
Linear Pair Postulate	Reason
Reflexive Property	Reason
\(m\angle ABC = 90^\circ\)	Statement
\(\angle A\) and \(\angle D\) are complementary angles	Statement
\(\angle 1 \cong \angle 2\)	Statement

</answer>

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

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"direct_cot",
"table_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Geometric Proofs",
"Mathematical Logic",
"Geometric Definitions",
"Postulates and Axioms"
],
"new_concepts": [],
"current_concepts": [
"Geometric Proofs",
"Mathematical Logic",
"Geometric Definitions",
"Postulates and Axioms"
]
}
</pre_analysis>

<reasoning>

Distinguish between statements and reasons in geometric proofs

A statement is a specific mathematical assertion or claim about a geometric figure (e.g., congruence, equality, bisecting relationships). A reason is a justification, definition, postulate, theorem, or property that supports why a statement is true.

Classify each given entry

• "Vertical Angles of Theorem" (likely a typo for "Vertical Angles Theorem"): Reason
• "\(\overleftrightarrow{PQ}\) bisects \(\overline{RS}\)": Statement
• "\(\overline{AB} \cong \overline{BC}\)": Statement
• "Given": Reason
• "Definition of Midpoint": Reason
• "Linear Pair Postulate": Reason
• "Reflexive Property": Reason
• "\(m\angle ABC = 90^\circ\)": Statement
• "\(\angle A\) and \(\angle D\) are complementary angles": Statement
• "\(\angle 1 \cong \angle 2\)": Statement

</reasoning>

<answer>

Entry	Classification
\(\overleftrightarrow{PQ}\) bisects \(\overline{RS}\)	Statement
\(\overline{AB} \cong \overline{BC}\)	Statement
Given	Reason
Definition of Midpoint	Reason
Linear Pair Postulate	Reason
Reflexive Property	Reason
\(m\angle ABC = 90^\circ\)	Statement
\(\angle A\) and \(\angle D\) are complementary angles	Statement
\(\angle 1 \cong \angle 2\)	Statement

</answer>

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